Author: shivamlohiya

Conic Section

A conic is the locus of a point whose distance from a fixed point bears a constant ratio to its distance from a fixed line. The fixed point is the focus S and the fixed line is directrix l.

The constant ratio is called the eccentricity denoted by e.

1. If 0 < e < 1, conic is an ellipse.
2. e = 1, conic is a parabola.
3. e > 1, conic is a hyperbola.
4. If fixed point of curve is (x₁, y₁) and fixed line is ax + by + c = then equation of the conic is
   (a² + b²) [(x — x₁)² + (y — y₁)²] = e²(ax + by + c)²

General Equation of Conic

A second degree equation ax² + 2hxy + by² + 2gx + 2fy + c= 0 represents

1. Pair of straight lines, if
2. Circle, if a = b, h = 0
3. Parabola, if h² = ab and Δ ≠ 0
4. Ellipse, if h² < ab and Δ ≠ 0
5. Hyperbola, if h² > ab and Δ ≠ 0
6. Rectangular hyperbola, if a + b = 0 and Δ ≠ 0

Parabola

A parabola is the locus of a point which moves in a plane such that its distance from a fixed point in the plane is always equal to its distance from a fixed straight line in the same plane.

If focus of a parabola is S(x₁, y₁) and equation of the directrix is ax + by + c = 0, then the equation of the parabola is

(a² + b²)[(x – x₁)² + (y – y₁)²] = (ax + by + c)²

Definitions Related to Parabola

1. Vertex The intersection point of parabola and axis.
2. Centre The point which bisects every chord of the conic passing through it.
3. Focal Chord Any chord passing through the focus.
4. Double Ordinate A chord perpendicular to the axis of a conic.
5. Latusrectum A double ordinate passing through the focus of the parabola.
6. Focal Distance The distance of a point P(x, y) from the focus S is called the focal distance of the point P.

Other Forms of a Parabola

If the vertex of the parabola is at a point A(h , k) and its latusrectum is of length 4a, then its equation is

1. (y – k)² = 4a (x – h), its axis is parallel to OX i. e. , parabola open rightward.
2. (y – k)² = – 4a (x – h), its axis is parallel to OX’ i. e., parabola open leftward.
3. (x – h)² = – 4a (y – k), its axis is parallel to OY i.e., parabola open upward.
4. (x – h)² = – 4a (y – k), its axis is parallel to OY ‘ i.e., parabola open downward. –
5. The general equation of a parabola whose axis is parallel to X – axis is x = ay² + by + c and the general equation of a parabola whose axis is parallel to Y-axis is y = ax² + bx + c.

Position of a Point

The point (x₁, y₁) lies outside, on or inside the parabola y² = 4ax according as y₁² — 4ax₁>, =, < 0.

Chord

Joining any two points on a curve is called chord.

(i) Parametric Equation of a Chord Let P(at₁² , 2at₁) and Q (at₂², 2at₂) be any two points on the parabola y₁ = 4ax, then the equation of the chord is

or y (t₁ + t₁) = 2x + 2at₁t₂

(ii) Let P(at² , 2at) be the one end of a focal chord PQ of the parabola y² = 4ax, then the coordinates of the other end Q are

(a/t², -2a/t)

(iii) If  l₁ and l₂ are the length of the focal segments, then length of the latusrectum = 2 (harmonic mean of focal segment)

i.e.,  

(iv) For a chord joining points P(at₁² , 2at₁) and Q(at₂² , 2at₂) and passing through focus, then t₁t₂ = 1.
(v) Length of the focal chord having t₁ and t₂ as end points is a (t₁ — t₁)².
(vi) Chord of contact drawn from a point (x₁, y₁) to the parabola y² = 4ax is yy₁, = 2a (x + x₁)

(vii) Equation of the chord of the parabola y² = 4ax, which is bisected at (x₁ , y₁) is given by

T = S₁
i.e. , yy₁ — 2a (x + x₁) = y₁² – 4ax

Equation of Tangent

A line which touch only one point of a parabola.

(i) Point Form The equation of the tangent to the parabola y² = 4ax at a point (x₁, y₁) is given by yy₁ = 2a (x + x₁)

(ii) Slope Form

(a) The equation of the tangent of slope m to the parabola y² = 4ax is

y = (mx + a/m)

(b) The equation of the tangent of slope m to the parabola (y – k)² = 4a (x – h) is given by

(y – k)² = m (x — h) + a/m

The coordinates of the point of contact are

(iii) Parametric Form The equation of the tangent to the parabola y² = 4ax at a point (at², 2at) is yt = x + at²

(iv) The line y = mx + c touches a parabola, if c = a/m and the point of contact is

(v) Point of Intersection of Two Tangents Let two tangents at P(at₁² , 2at₁) and Q(at₂², 2at₂) intersect at R. Then, their point of intersection is R[at₁t₂, a(t₁ + t₂)] i.e., (GM of abscissa, AM of ordinate).

(vi) The straight line lx + my + n = 0 touches y² = 4ax, if nl = am² and x cos α + y sin α = p touches y² = 4ax, if p cos α + a sin² α = 0.

(vii) Angle θ between tangents at two points P(at₁t₂ , 2at₁) and Q(at₂², 2at₂) on the parabola y² = 4ax is given by

(viii) The combined equation of the pair of tangents drawn from a point to a parabola y² = 4ax is given by

SS₁ = T²

where, S = y² – 4ax, S₁= y₁² – 4ax₁

and T = [yy₁ – 2a (x + x₁)]

Important Results on Tangents

1. The tangent at any point on a parabola bisects the angle between the focal distance of the point and the perpendicular on the directrix from the point.
2. The tangent at the extremities of a focal chord of a parabola intersect at right angle on the directrix.
3. The portion of the tangent to a parabola cut off between the directrix and the curve subtends a right angle at the focus.
4. The perpendicular drawn from the focus on any tangent to a parabola intersect it at the point where it cuts the tangent at the vertex.
5. The orthocentre of any triangle formed by three tangents to a parabola lies on the directrix.
6. The circumcircle formed by the intersection points of tangents at any three points on a parabola passes through the focus of the parabola.
7. The tangent at any point of a parabola is equally inclined to the focal distance of the point and the axis of the parabola.
8. The length of the subtangent at any point on a parabola is equal to twice the abscissa of the point.
9. Two tangents can be drawn from a point to a parabola. Two tangents are real and distinct or coincident or imaginary according as given point lies outside, on or inside the parabola.

Equation of Normal

A line which is perpendicular to the tangent.

1. Point Form The equation of the normal to the parabola y² = 4ax at a point (x₁, y₁) is given by y – y₁ = -y₁/2a(x — x₁).
2. Parametric Form The equation of the normal to the parabola y² = 4ax at point (at², 2at) is given by y + tx = 2at + at³
3. Slope Form The equation of the normal to the parabola y² = 4ax in terms of its slope m is given by y = mx — 2am — am³ at point (am², — 2am).

Important Results on Normals

1. If the normal at the point P(at₁²,2at₁) meets the parabola y² = 4ax at Q(a₂²,2at₂), then t₂ = -t – 2/t₁.
2. The tangent at one extremity of the focal chord of a parabola is parallel to the normal at other extremity.
3. The normal at points P(at₁²,2at₁) and Q(a₂²,2at₂) to the parabola y² = 4ax intersect at the point
   [2a + a(t₁² + t₂²+ t₁t₂)], – at₁t₂(t₁ + t₂).
4. If the normal at points P(at₁²,2at₁) and Q(a₂²,2at₂) on the parabola y² = 4ax meet on the parabola, then t₁t₂ = 2.
5. If the normal at two points P and Q of a parabola y² = 4ax intersect at a third point R on the curve, then the product of the ordinates of P and Q is 8a².
6. If the normal chord at a point P(at² ,2at)to the parabola y² = 4ax subtends a right angle at the vertex of the parabola, then t² = 2.
7. The normal chord of a parabola at a point whose ordinate is equal to the abscissa, subtends a right angle at the focus.
8. The normal at any point of a parabola is equally inclined to the focal distance of the point and the axis of the parabola.
9. Three normals can be drawn from a point to a parabola.
10. Conormai The points on the parabola at which the normals pass through a common point are called conormal points. The conormal points are called the feet of the normals.

Points A, B and Care called conormal points.

• The algebraic sum of the slopes of the normals at conormals point is O.
• The sum of the ordinates of the conormal points is O.
• The centroid of the triangle formed by the conormal points on a parabola lies on its axis.

Director Circle

• The locus of the point of intersection of perpendicular tangents to a conic is known as director circle.
• The director circle of a parabola is its directrix.

Equation of Diameter

• The locus of mid-point of a system of parallel chords of a conic is known its diameter.
• The diameter bisecting chords of slope m to the parabola y² = 4ax is y = 2a/m

Length of Tangent and Normal

1. The length of the tangent = PT = PN cosec Ψ = y₁ cosec Ψ
2. The length of subtangent = NT = PN cot Ψ = y₁ cot Ψ
3. The length of normal = PG = PN sec Ψ = y₁ sec Ψ
4. The length of subnormal = NG = PN tan Ψ = y₁ tan Ψ

Pole and Polar

Let P be a point lying within or outside a given parabola. Suppose any straight line drawn through P intersects the parabola at Q and R. Then, the locus of the point of intersection of the tangents to the parabola at Q and R is called the polar of the given point P with respect to the parabola and the point P is called the pole of the polar.

1. The polar of a point P(x₁, ₁) with respect to the parabola y² = 4ax is yy₁ = 2a(x + x₁) or T = 0.
2. Any tangent is the polar of its point of contact.
3.  Pole of lx + my + n = 0 with respect to y² = 4ax is
4. Pole of the chord joining (x₁, y₁) and (x₁, y₁) is
5. If the polar of P(x₁, y₁)passes through Q(x₂ ,y₂), then the polar of Q will passes through P. Here, P and Q are called conjugate points.
6. If the pole of a line a₁x + b₁y + c₁ = 0 lies on another line a₂x + b₂y + c₂ = 0, then the pole of the second line will lies on the first line. Such lines are called conjugate lines.
7. The point of intersection of the polar of two points Q and R is the pole of QR.
8. The tangents at the ends of any chord of the parabola meet on the diameter which bisect the chord.

Important Points to be Remembered

(i) For the ends of latusrectum of the parabola y² = 4ax, the values of the perimeter are ± 1.
(ii) The circles described on focal radii of a parabola as diameter touches the tangent at the vertex.
(iii) The straight line y =mx + c meets the parabola y² = 4ax in two points. These two points are real and distinct, if c > a/m, points are real and coincident, if c = a/m, points are imaginary, if c < a/m.
(iv) Area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points.
(v) The circles described on any focal chord of a parabola as diameter touches the directrix.
(vi) If y₁, y₂, y₃ are the ordinates of the vertices of a triangle inscribed in the parabola y² = 4ax, then its area is 1/8a (y₁ – y₂) (y₂ – y₃) (y₃ – y₁).

Circle is defined as the locus of a point which moves in a plane such that its distance from a fixed point in that plane is constant.

Standard Forms of a Circle

(i) Equation of circle having centre (h, k) and radius (x — h)² + (y — k)² = a².

If centre is (0, 0), then equation of circle is x² + y² = a².

(ii) When the circle passes through the origin, then equation of the circle is x² + y² — 2hx — 2ky = 0.

(iii) When the circle touches the X-axis, the equation is x² + y² — 2hx — 2ay + h² = O.

(iv) Equation of the circle, touching the Y-axis is x² + y² — 2ax — 2ky + k² = 0.

(v) Equation of the circle, touching both axes is x² + y² — 2ax — 2ay + a² = O.

(vi) Equation of the circle passing through the origin and centre lying on the X-axis is x² + y² — 2ax = O.

(vii) Equation of the circle passing through the origin and centre lying on the Y-axis is x² + y² – 2ay = 0.

(viii) Equation of the circle through the origin and cutting intercepts a and b on the coordinate axes is x² + y² — by = 0.

(ix) Equation of the circle, when the coordinates of end points of a diameter are (x₁, y₁) and (x₂, y₂) is

(x — x₁)(x — x₂) + (y – y₁)(y — y₂) = 0.

(x) Equation of the circle passes through three given points (x₁, y₁), (x₂, y₂) and (x₃, y₃) is

(xi) Parametric equation of a circle

(x – h)² + (y – k)² = a² is
x = h + a cosθ, y = k + a sinθ,
0 ≤ θ ≤ 2π
For circle x² + y² = a², parametric equation is
x = a cos θ, y = a sin θ

General Equation of a Circle

The general equation of a circle is given by x² + y² + 2gx + 2fy + c = 0, where centre of the circle = (- g, – f)

Radius of the circle = √g² + f² – c

1. If g² + f² – c > 0, then the radius of the circle is real and hence the circle is also real.
2. If g² + f² – c = 0, then the radius of the circle is 0 and the circle is known as point circle.
3. If g² + f² – c< 0, then the radius of the circle is imaginary. Such a circle is imaginary, which is not possible to draw.

Position of a Point with Respect to a Circle

A point (x₁, y₁) lies outside on or inside a circle

S ≡ x² + y² + 2gx + 2fy + c = 0, according as S₁ > , = or < 0
where, S₁ = x₁² + y₁² + 2gx₁, + 2fy₁+ c

Intercepts on the Axes

The length of the intercepts made by the circle x² + y² + 2gx + 2fy + c = 0 with X and Y-axes are

2√g² – c and 2√g² – c.

1. If g² > c, then the roots of the equation x² + 2gx + c = 0 are real and distinct, so the circle x² + y² + 2gx + 2fy + c = 0 meets the X-axis in two real and distinct points.
2. If g² = c, then the roots of the equation x² + 2gx + c = 0 are real and equal, so the circle touches X-axis, then intercept on X-axis is O.
3. If g² < c, then the roots of the equation x² + 2gx + c = 0 are imaginary, so the given circle does not meet X-axis in real point. Similarly, the circle x² + y² + 2gx + 2fy + c = 0 cuts the Y-axis in real and distinct points touches or does not meet in real point according to f² >, = or < c

Equation of Tangent

A line which touch only one point of a circle.

1. Point Form

1. The equation of the tangent at the point P(x₁, y₁) to a circle x² + y² 2gx + 2fy + c= 0 is xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0
2. The equation of the tangent at the point P(x₁, y₁) to a circle x² + y² is xx₁ + yy₁ = r²

2. Slope Form

(i) The equation of the tangent of slope m to the circle x² + y² + 2gx + 2fy + c = 0

are y + f = m(x + g) ± √(g² + f² — c)(1 + m²)

(ii) The equation of the tangents of slope m to the circle (x – a)² (y – b)² = r² are y – b = m(x – a) ± r√(1 + m²) and the coordinates of the points of contact are

(iii) The equation of tangents of slope m to the circle x² + y² = r² are y = mx ± r√(1 + m²)and the coordinates of the point of contact are

3. Parametric Form

The equation of the tangent to the circle (x – a)² + (y – b)² = r² at the point (a + r cos θ, b + r sinθ) is (x – a) cos θ + (y – b) sin θ = r.

Equation of Normal

A line which is perpendicular to the tangent.

1. Point Form

1. (i) The equation of normal at the point (x₁, y₁) to the circle x² + y² + 2gx + 2fy + c = 0 is
   y – y₁ = [(y₁ + f)(x – x₁)]/(x₁ + g)
   (y₁ + f)x – (x₁ + g)y + (gy₁ – fx₁) = 0
2. (ii) The equation of normal at the point (x₁, y₁) to the circle
   x² + y² = r² is x/x₁ = y/y₁

2. Parametric Form

The equation of normal to the circle x² + y² = r² at the point (r cos θ, r sin θ) is

(x/r cos θ) = (y/r sin θ)

or y = x tan θ.

Important Points to be Remembered

(i) The line y = mx + c meets the circle in unique real point or touch the circle

x₂ + y₂ + r₂, if r = |c/√1 + m₂

and the point of contacts are

(ii) The line lx + my + n = 0 touches the circle x₂ + y₂ = r₂, if r₂(l₂ + m₂) = n₂.

(iii) Tangent at the point P (θ) to the circle x₂ + y₂ = r₂ is x cos θ + y sin θ = r.

(iv) The point of intersection of the tangent at the points P(θ₁) and Q(θ₂) on the circle x² + y² = r²

(v) Normal at any point on the circle is a straight line which is perpendicular to the tangent to the curve at the point and it passes through the centre of circle.

(vi) Power of a point (x₁, y₁) with respect to the circle x² + y² + 2gx + 2fy + c = 0 is x₁² + y₁² + 2gx₁ + 2fy₁ + c.

(vii) If P is a point and C is the centre of a circle of radius r, then the maximum and minimum distances of P from the circle are CP + r and CP — r , respectively.

(viii) If a line is perpendicular to the radius of a circle at its end points on the circle, then the line is a tangent to the circle and vice-versa.

Pair of Tangents

(i) The combined equation of the pair of tangents drawn from a point P(x₁, y₁) to the circle x² + y² = r² is

(x² + y² – r²)(x₁²+ y₁² – r₁²) = (xx₁ + yy₁ – r²)²
or SS₁ = T²
where, S = x² + y² – r², S₁ = x₁²+ y₁² – r₁²
and T = xx₁ + yy₁ – r²

(ii) The length of the tangents from the point P(x₁, y₁) to the circle x² + y² + 2gx + 2fy + c = 0 is equal to

(iii) Chord of contact TT’ of two tangents, drawn from P(x₁, y₁) to the circle x² + y² = r² or T = 0.

Similarly, for the circle

x² + y² + 2gx + 2fy + c = 0 is

xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0

(iv) Equation of Chord Bisected at a Given Point The equation of chord of the circle S ≡ x² + y² + 2gx + 2fy + c = 0 bisected at the point (x₁, y₁) is give by T = S₁.
i.e., xx₁ + yy₁ + g (x + x₁) + f (y + y₁) + c
= x₁² + y₁² + 2gx₁ + fy₁ + c

(v) Director Circle The locus of the point of intersection of two perpendicular tangents to a given circle is called a director circle. For circle x² + y² = r², the equation of director circle is x² + y² = 2r².

Common Chord

The chord joining the points of intersection of two given circles is called common chord.

(i) If S₁ = 0 and S₁ = 0 be two circles, such that

S₁ ≡ x² + y² + 2g₁x + 2f₁y + c₁ = 0
and S₂ ≡ x² + y² + 2g₂x + 2f₂y + c₂ = 0
then their common chord is given by S₁ — S₂ = 0

(ii) If C₁, C₂ denote the centre of the given circles, then their common chord

PQ = 2 PM = 2√(C₁P)² – C₁M)²

(iii) If r₁, and r₂ be the radii of ‘two circles, then length of common chord is

Angle of Intersection of Two Circles

The angle of intersection of two circles is defined as the angle between the tangents to the two circles at their point of intersection is given by

cos θ = (r₁² + r₂² – d²)/(2r₁r₂)

Orthogonal Circles

Two circles are said to be intersect orthogonally, if their angle of intersection is a right angle.

If two circles

S₁ ≡ x² + y² + 2g₁x + 2f₁y + C₁ = 0 and

S₂ ≡ x² + y² + 2g₂x + 2f₂y + C₂ = 0 are orthogonal, then 2g₁g₂ + 2f₁f₂ = c₁ + c₂

Family of Circles

(i) The equation of a family of circles passing through the intersection of a circle x² + y² + 2gx + 2fy + c = 0 and line

L = lx + my + n = 0 is S + λL = 0

where, X, is any real number.

(ii) The equation of the family of circles passing through the point A(x₁, y₁) and B (x₁, y₁) is

(iii) The equation of the family of -circles touching the circle

S ≡ x² + y² + 2gx + 2fy + c = 0 at point P(x₁, Y₁) is

xx² + y² + 2gx + 2fy + c + λ, [xx₁ + yy₁ + g(x + x₁) + f(Y+ Y₁) + c] = 0 or S + λL = 0, where L = 0 is the equation of the tangent to

S = 0 at (x₁, y₁) and X ∈ R

(iv) Any circle passing through the point of intersection of two circles S₁ and S₂ is S₁ +λ(S₁— S₂) = 0.

Radical Axis

The radical axis of two circles is the locus of a point which moves in such a way that the length of the tangents drawn from it to the two circles are equal.

A system of circles in which every pair has the same radical axis is called a coaxial system of circles.

The radical axis of two circles S₁ = 0 and S₂ = 0 is given by S₁ — S₂ = 0.

1. The radical axis of two circles is always perpendicular to the line joining the centres of the circles.
2. The radical axis of three vertices, whose centres are non-collinear taken in pairs of concurrent.
3. The centre of the circle cutting two given circles orthogonally, lies on their radical axis.
4. Radical Centre The point of intersection of radical axis of three circles whose centre are non-collinear, taken in pairs, is called their radical centre.

Pole and Polar

If through a point P (x₁, y₁) (within or outside a circle) there be drawn any straight line to meet the given circle at Q and R, the locus of the point of intersection of tangents at Q and R is called the polar of P and po.:.at P is called the pole of polar.

1. Equation of polar to the circle x² + y² = r² is xx₁ + yy₁ = r².
2. Equation of polar to the circle x² + y² + 2gx + 2fy + c = 0 is xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0
3. Conjugate Points Two points A and B are conjugate points with respect to a given circle, if each lies on the polar of the other with respect to the circle.
4. Conjugate Lines If two lines be such that the pole of one lies on the other, then they are called conjugate lines with respect to the given circle.

Coaxial System of Circles

A system of circle is said to be coaxial system of circles, if every pair of the circles in the system has same radical axis.

1. The equation of a system of coaxial circles, when the equation of the radical axis P ≡ lx + my + n = 0 and one of the circle of the system S = x² + y² + 2gx + 2fy + c = 0, is S + λP = 0.
2. Since, the lines joining the centres of two circles is perpendicular to their radical axis. Therefore, the centres of all circles of a coaxial system lie on a straight line, which is perpendicular to the common radical axis.

Limiting Points

Limiting points of a system of coaxial circles are the centres of the point circles belonging to the family.

Let equation of circle be x² + y² + 2gx + c = 0

∴ Radius of circle = √g² — c

For limiting point, r = 0

∴ √g² — c = 0 &rArr;g = ± √c

Thus, limiting points of the given coaxial system as (√c, 0) and (—√c, 0).

Important Points to be Remembered

(i) Circle touching a line L=O at a point (x₁, y₁) on it is

(x — x₁)² + (y — y₁)² + XL = 0.

(ii) Circumcircle of a A with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is

(iii) A line intersect a given circle at two distinct real points, if the length of the perpendicular from the centre is less than the radius of the circle.

(iv) Length of the intercept cut off from the line y = mx + c by the circle x² + y² = a² is

(v) In general, two tangents can be drawn to a circle from a given point in its plane. If m₁and m₂ are slope of the tangents drawn from the point P(x₁, y₁) to the circle x² + y² = a², then

(vi) Pole of lx + my + n = 0 with respect to x² + y² = a² is

(vii) Let S₁ = 0, S₂ = 0 be two circles with radii r₁ , r₂, then S₁/r₁ ± S₂/r₂ = 0 will meet at right angle.

(viii) The angle between the two tangents from (x₁, y₁) to the circle x² + y² = a² is 2 tan^-1 (a/√S₁).

(ix) The pair of tangents from (0, 0) to the circle x² + y² + 2gx + 2fy + c = 0 are at right angle, if g² + f² = 2c.

(x) If (x₁, y₁) is one end of a diameter of the circle x² + y² + 2gx + 2fy + c = 0, then the other end will be (-2g – x₁, -2f – y₁).

Image of the Circle by the Line Minor

Let the circle be x² + y² + 2gx + 2fy + c = 0

and line minor lx + my + n = 0.

Then, the image of the circle is

(x — X₁)² + (y — y₁)² =r²

where, r = √g² + f² — c

Diameter of a Circle

The locus of the middle points of a system of parallel chords of a circle is called a diameter of the circle.

(i) The equation of the diameter bisecting parallel chords y = mx + c of the circle x² + y² = a² is x + my = 0.

(ii) The diameter corresponding to a system of parallel chords of a circle always passes through the centre of the circle and is perpendicular to the parallel chords.

Common Tangents of Two Circles

Let the centres and radii of two circles are C₁, C₂ and r₁, r₂, respectively.

1. (i) When one circle contains another circle, no common tangent is possible.
   Condition, C₁C₂ < r₁ – r₂
2. (ii) When two circles touch internally, one common tangent is possible.
   Condition , C₁C₂ = r₁ – r₂
3. (iii) When two circles intersect, two common tangents are possible.
   Condition, |r₁ — r₂| < C₁C₂ < |r₁ + r₂|
4. (iv) When two circles touch externally, three common tangents are possible.
   Condition, C₁C₂ = r₁ + r₂
5. (v) When two circles are separately, four common tangents are possible.
   Condition, C₁C₂ > r₁ + r₂

Important Points to be Remembered

Let AS is a chord of contact of tangents from C to the circle x² + y² = r². M is the mid-point of AB.

Coordinate Geometry

The branch of Mathematics in which geometrical problem are solved through algebra by using the coordinate system, is known as coordinate geometry.

Rectangular Axis

Let XOX’ and YOY’ be two fixed straight lines, which meet at right angles at O. Then,

(i) X’OX is called axis of X or the X-axis or abscissa.

(ii) Y’OY is called axis of Yor the Y-axis or ordinate.

(iii) The ordered pair of real numbers (x, y) is called cartesian coordinate .

Quadrants

The X and Y-axes divide the coordinate plane into four parts, each part is called a quadrant which is given below.

Polar Coordinates

In ΔOPQ,

cos θ = x / r and sin θ = y / r ⇒ x = r cos θ and y = r sin θ

where, r = √x² + y²

The polar coordinate is represented by the symbol P(r,θ).

Distance Formula

(i) Distance between two points P (x₁, y₁) and Q (x₂, y₂), is

√(x₂ – x₁)² + (y₂ – y₁)².

(ii) If points are (r₁ , θ₁) arid (r₂, θ₂), then distance between them is

√r²₁ + r²₂ – 2r₁r₂cos(θ₁ – θ₂).

(iii) Distance of a point (x₁, y₁) from the origin is √x²₁ + y²₁.

Section Formula

(i) The coordinate of the point which divides the joint of (x₁, y₁) and (x₂, y₂) in the ratio m₁: m₂ internally, is

(ii) X-axis divides the line segment joining (x₁, y₁) and (x₂, y₂) in the ratio – y_{1 : y2.}

Similarly, Y-axis divides the same line segment in the ratio – x_{1 : x2.}

(iii) Mid-point of the joint of (x₁, y₁) and (x₂, y₂) is (x₁ + x₂ / 2 , y₁ + y₂ / 2)

(iv) Centroid of ΔABC with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃), is

(x₁ + x₂ + x₃ / 3 , y₁ + y₂ + y₃ / 3).

(v) Circumcentre of ΔABC with vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃), is

(vi) Incentre of Δ ABC with vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) and whose sides are a, band c, is

(ax₁ + bx₂ + cx₃ / a + b + c , ay₁ + by₂ + cy₃ / a + b + c).

(vii) Excentre of ΔABC with vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) and whose sides are a, band c, is given by

Area of Triangle/Quadrilateral

(i) Area of ΔABC with vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃), is

These points A, Band C will be collinear, if Δ = O.

(ii) Area of the quadrilateral formed by joining the vertices

(iii) Area of trapezium formed by joining the vertices

Shifting/Rotation of Origin/Axes

Shifting of Origin

Let the origin is shifted to a point O'(h, k). If P(x, y) are coordinates of a point referred to old axes and P’ (X, Y) are the coordinates of the same points referred to new axes, then

Rotation of Axes

Let (x, y) be the coordinates of any point P referred to the old axes and (X, Y) be its coordinates referred to the new axes (after rotating the old axes by angle θ). Then,

X = x cos θ + y sin θ and Y = y cos θ + x sin θ

Shifting of Origin and Rotation of Axes

If origin is shifted to point (h, k) and system is also rotated by an angle θ in anti-clockwise, then coordinate of new point P’ (x’, y’) is obtained by replacing

x’= h + x cos θ + y sin θ

and y’ = k – x sin θ + y cos θ

Locus

The curve described by a point which moves under given condition(s) is called its locus.

Equation of Locus

The equation of the locus of a point which is satisfied by the coordinates of every point.

Algorithm to Find the Locus of a Point

Step I Assume the coordinates of the point say (h,k) whose locus is to be found.

Step II Write the given condition in mathematical form involving h, k.

Step III Eliminate the variable(s), if any.

Step IV Replace h by x and k by y in the result obtained in step III. The equation so obtained is the locus of the point, which moves under some stated condition(s).

Straight Line

Any curve is said to be a straight line, if two points are taken on the curve such that every point on the line segment joining any two points on it lies on the curve.

General equation of a line is ax + by + c = o.

Slope (Gradient) of a Line )

The trigonometric tangent of the angle that a line makes with the positive direction of the X-axis in anti-clockwise sense is called the slope or gradient of the line.

So, slope of a line, m = tan θ

where, θ is the angle made by the line with positive direction of X-axis.

Important Results on Slope of Line

(i) Slope of a line parallel to X-axis, m = 0.
(ii) Slope of a line parallel to Y-axis, m = ∞.
(iii) Slope of a line equally inclined with axes is 1 or -1 as it makes an angle of 45° or 135°, with X-axis.
(iV) Slope of a line passing through (x, y,) and (x₂, y₂) is given by

m = tan θ = y₂ – y₁ / x₂ – x₁.

Angle between Two Lines

The angle e between two lines having slopes m₁ and m₂ is

(i) Two lines are parallel, iff m₁ = m₂.
(ii) Two lines are perpendicular to each other, iff m₁m₂ = – 1.

Equation of a Straight Line

General equation of a straight line is Ax + By + C = 0.

(i) The equation of a line parallel to X-axis at a distance b from it, is given by

y = b

(ii) The equation of a line parallel to Y-axis at a distance a from it, is given by

x = a

(iii) Equation of X-axis is

y = 0

(iv) Equation of Y-axis is

x = 0

Different Form of the Equation of a Straight Line

(i) Slope Intercept Form The equation of a line with slope m and making an intercept c on Y-axis, is

y = mx + c

If the line passes through the origin, then its equation will be

y= mx

(ii) One Point Slope Form The equation of a line which passes through the point (x₁, y₁) and has the slope of m is given by

(y – y₁) = m (x – x₁)

(iii) Two Points Form The equation of a line’ passing through the points (x₁, y₁) and (x₂, y₂) is given by

(y – y₁) = (y₂ – y₁ / x₂ – x₁) (x – x₁)

This equation can also be determined by the determinant method, that is

(iv) The Intercept Form The equation of a line which cuts off intercept a and b respectively on the X and Y-axes is given by

x / a + y / b = 1

The general equation Ax + By + C = 0 can be converted into the intercept form, as

x / – (C A) + y / – (C B) = 1

(v) The Normal Form The equation of a straight line upon which the length of the perpendicular from the origin is p and angle made by this perpendicular to the X-axis is α, is given by

x cos α + Y sin α = p

(vi) The Distance (Parametric) Form The equation of a straight line passing through (x₁, y₁) and making an angle θ with the positive direction of x-axis, is

x – x₁ / cos θ = y – y₁ / sin θ = r

where, r is the distance between two points P(x, y) and Q(x₁, y₁).

Thus, the coordinates of any point on the line at a distance r from the given point (x₁, y₁) are (x₁ + r cos θ, y₁ + r sin θ). If P is on the right side of (x₁, y₁) then r is positive and if P is on the left side of (x₁, y₁) then r is negative.

Position of Point(s) Relative to a Given Line

Let the equation of the given line be ax + by + C = 0 and let the Coordinates of the two given points be P(x₁, y₁) and Q(x₂, y₂).

(i) The two points are on the same side of the straight line ax + by + c = 0, if ax₁ + by₁ + c and ax₂ + by₂ + c have the same sign.

(ii) The two points are on the opposite side of the straight line ax + by + c = 0, if ax₁ + by₁ + c and ax₂ + by₂ + c have opposite sign.

(iii) A point (x₁, y₁) will lie on the side of the origin relative to a line ax + by + c = 0, if ax₁+ by₁ + c and c have the same sign.

(iv) A point (x₁, y₁) will lie on the opposite side of the origin relative to a line ax + by + c = 0, if ax₁ + by₁ + c and c have the opposite sign.

(v) Condition of concurrency for three given lines ax₁ + by₁ + c₁ = 0, ax₂ + by₂ + c₂ and ax₃ + by₃ + c₃ = 0 is a₃(b₁c₂ – b₂c₁) + b₃(c₁a₂ – a₁c₂) + c₃(a₁b₂ – a₂b₁) = 0

or

(vi) Point of Intersection of Two Lines Let equation of lines be ax₁ + by₁ + c₁ = 0 and ax₂ + by₂ + c₂ = 0, then their point of intersection is

(b₁c₂ – b₂c₁ / a₁b₂ – a₂b₁, c₁a₂ – c₂a₁ / a₁b₂ – a₂b₁).

Line Parallel and Perpendicular to a Given Line

(i) The equation of a line parallel to a given line ax + by + c = 0 is ax + by + λ = 0, where λ is a constant.

(ii) The equation of a line perpendicular to a given line ax + by + c = is bx – ay + λ = 0, where λ is a constant.

Image of a Point with Respect to a Line

Let the image of a point (x₁, y₁) with respect to ax + by + c = 0 be (x₂, y₂), then

x₂ – x₁ / a = y₂ – y₁ / b = – 2 (ax₁ + by₁ + c) / a² + b²

(i) The image of the point P(x₁, y₁) with respect to X-axis is Q(x₁ – y₁).

(ii) The image of the point P(x₁, y₁) with respect to Y-axis is Q(-x₁, y₁).

(iii) The image of the point P(x₁, y₁) with respect to mirror Y = x is Q(y₁, x₁).

(iv) The image. of the point P(x₁, y₁) with respect to the line mirror y == x tan θ is

x = x₁ cos 2θ + y₁ sin 2θ
Y = x₁ sin 2θ – y₁ cos 2θ

(v) The image of the point P(x₁, y₁) with respect to the origin is the point (-x₁, y₁).

(vi) The length of perpendicular from a point (x₁, y₁) to a line ax + by + c = 0 is

Equation of the Bisectors

The equation of the bisectors of the angle between the lines

a₁x + b₁y + c₁ = 0

and a₂x + b₂y + c₂ = 0

are given by

a₁x + b₁y + c₁ / √a²₁ + b²₁ = &plusmn a₂x + b₂y + c₂ / √a²₂ + b²₂

(i) If a₁ a₂ + b₁ b₂ > 0, then we take positive sign for obtuse and negative sign for acute.

(ii) If a₁ a₂ + b₁ b₂ < 0, then we take negative sign for obtuse and positive sign for acute .

Pair of Lines

General equation of a pair of straight lines is ax² + 2hxy + by² + 2gx + 2fy + c = 0.

Homogeneous Equation of Second Degree

A rational, integral, algebraic equation in two variables x and y is said to be a homogeneous equation of the second degree, if the sum of the indices of x and y in each term is equal to 2.

The general form of homogeneous equation of the second degree x and y is ax² + 2hxy + by² = 0, which passes through the origin.

Important Properties

If ax² + 2hxy + by² = 0 be an equation of pair of straight lines.

(i) Slope of first line, m₁ = – h + √h² – ab / b

and slope of second line, m₂ = – h – √h² – ab / b

m₁ + m₂ = – 2h / b = – Coefficient of xy / Coefficient of y²

and m₁ m₂ = a / b = Coefficient of x² / Coefficient of y²

Here, m₁ and m₂ are

(a) real and distinct, if h² > ab.

(b) coincident, if h² = ab.

(c) imaginary, if h² < ab.

(ii) Angle between the pair of lines is given by

tan θ = ²√h² – ab / a + b

(a) If lines are coincident, then h² = ab
(b) If lines are perpendicular, then a + b = o.

(iii) The joint equation of bisector of the angles between the lines represented by the equation ax² + 2hxy + by² = 0 is

x² – b² / a – b = xy / h ⇒ hx² – (a – b)xy – hy²=0.

(iv) The necessary and sufficient condition ax² + 2hxy + by² + 2gx + 2fy + C = 0 to represent a pair of straight lines, if abc + 2fgh – af² – bg² – ch² = 0

or

(v) The equation of the bisectors of the angles between the lines represented by ax² + 2hxy + by² + 2gx + 2fy + c = 0 are given by

(x – x¹)² – (y – y¹)² / a – b = (x – x¹) (y – y¹) / h,

where, (x¹, y¹) is the point of intersection of the lines represented by the given equation.

(vi) The general equation ax² + 2hxy + by² + 2gx + 2fy + C = 0 will represent two parallel lines, if g² – ac > 0 and a / h = h / b = g / f and the distance between them is 2√g² – ac / a(a + b) or 2√f² – bc / b(a + b).

(vii) If the equation of a pair of straight lines is ax² + 2hxy + by² + 2gx + 2fy + C = 0, then the point of intersection is given by

(hf – bg / ab – h², gh – af / ab – h²).

(viii) The equation of the pair of lines through the origin and perpendicular to the pair of lines given by ax² + 2hxy + by² = 0 is bx² – 2hxy + ay² = 0.

(ix) Equation of the straight lines having the origin to the points of intersection of a second degree curve ax² + 2hxy + by² + 2gx + 2fy + c = 0 and a straight line Lx + my + n = 0 is

ax² + 2hxy + by² + 2gx(Lx + my / – n) + 2fy(Lx + my / – n) + c (Lx + my / – n)² = 0.

Important Points to be Remembered

1. A triangle is an isosceles, if any two of its median are equal.

2. In an equilateral triangle, orthocentre, centroid, circumcentre, incentre coincide.

3. The circumcentre of a right angled triangle is the mid-point of the hypotenuse.

4. Orthocentre, centroid, circumcentre of a triangle are collinear, Centroid divides the line joining orthocentre and circumcentre in the ratio 2: 1.

5. If D, E and F are the mid-point of the sides BC, CA and AB of MBC, then the centroid of Δ ABC = centroid of Δ DEF.

6. Orthocentre of the right angled Δ ABC, right angled at A is A

7. Circumcentre of the right angled Δ ABC, right angled at A is B + C / 2.

8. The distance of a point (x¹, y¹) from the ax + by + c = 0 is

d = |ax₁ + by₁ + c / √a² + b²|

9. Distance between two parallel lines a₁x + b₁y + c₁ = 0 and a₁x + b₁y + c₂ = 0 is given by

d = |c₂ – c₁ / √a² + b²|.

10. The area of the triangle formed by the lines y =m₁x + c₁, y = m₂x + c₂ and y = m₃x + c₃ is .

.

11. Area of the triangle formed by the line ax + by + c = 0 with the coordinate axes is Δ = c² / 2|ab|.

12. The foot of the perpendicular(h,k) from (x₁, y₁) to the line ax + by + c = 0 is given by h – x₁ / a = k – y₁ / b = – (ax₁ + by₁ + c) / a² + b².

13. Area of rhombus formed by ax ± by ± c = 0 is |2c² / ab|.

14. Area of the parallelogram formed by the lines

a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0, a₁x + b₁y + d₁ = 0 and a₂x + b₂y + d₂ = 0 is

|(d₁ – c₁) (d₂ – c₂ / a₁b₂ – a₂b₁|.

15. (a) Foot of the perpendicular from (a, b) on x – y = 0 is

(a + b / 2, a + b / 2).

(b) Foot of the perpendicular from (a,b) on x + y = 0 is

(a – b / 2, a – b / 2).

16. The image of the line a₁x + b₁y + c₁ = 0 about the line ax + by + c = 0 is .

2(aa₁ + bb₁) (ax + by + c) =(a²+ b²) (a₁x + b₁y + c₁).

17. Given two vertices (x₁, y₁) and (x₂, y₂) of an equilateral MBC, then its third vertex is given by.

[x₁ + x₂ ± √3 (y₁ – y₂) / 2, y₁ + y₂ ∓ √3 (x₁ – x₂) / 2]

18. The equation of the straight line which passes through a given point (x₁, y₁) and makes an angle α with the given straight line y = mx + c are

19. The equation of the family of lines passing through the intersection of the lines a₁x + b₁y+ c₁ = 0 and a₂x + b₂y + c₂ = 0 is

(a₁x + b₁y+ c₁) + l(a₂x + b₂y + c₂) = 0

where, λ is any real number.

20. Line ax + by + c = 0 divides the line joining the points (x₁, y₁) and (x₂, y₂) in the ratio λ : 1, then λ = – (a₁x + b₁y+ c / a₂x + b₂y + c).

If λ is positive it divides internally and if λ. is negative, then it divides externally.

21. Area of a polygon of n-sides with vertices A₁(x₁, y₁), A₂ (x₂, y₂) ,… ,A_n(x_n, y_n)

22. Equation of the pair of lines through (α, β) and perpendicular to the pair of lines ax² + 2hxy + by² = 0 is b (x – α}² – 2h (x – α)(y – β) + a (y – β)²= 0.

Basic Rules of Triangle

In ΔABC, the angles are denoted by capital letters A, Band C and the lengths of the sides opposite to these angles are denoted by small letters a, b and c, respectively. Semi-perimeter of the triangle is written as

s= a + b + c / 2

and area denoted by Δ.

(i) Sine Rule sin A / a = sin B / b = sin C / c = 1 / 2R, where R is the radius of the circumstances of the ΔABC.

(ii) Cosine Rule

cos A = b² + c² – a² / 2bc

cos B = a² + c² – b² / 2ac

cos C = a² + b² – c² / 2ab

(iii) Projection Rule a = b cos C + c cos B,

b = c cos A + a cos C and c = a cos B + b cos A

(iV) Napier’s Analogy tan B – C / 2 = b – c / b + c cot A / 2,

tan C – A / 2 = c – a / c + a cot B / 2 and tan A – B / 2 = a – b / a + b cot C / 2

Trigonometrical Ratios of Half of the Angles

Area of Triangles

Consider a triangle of side a, b and c.

(iv) Δ = abc / 4R = rs

where, R and r are the radii of the circumcircle and the incircle of the ΔABC, respectively.

Solution of Triangles

Elements of a Triangle

There are six elements of a triangle, in which three are its sides and other three are its angle.

If three elements of a triangle are given, atleast one of which is its side; then other elements can be uniquely calculated. This is called solving the triangle.

1. Solutions of a Right Angled Triangle

Let ΔABC be a given triangle with right angle at C, then

(i) solution when two sides are given

(ii) solution when one side and one a cute angle are given

2. Solutions of a Triangle in General

(i) Solution when three sides a, band c are given, then

sin A = 2Δ / bc, sin B = 2Δ / ac, sin C = 2Δ / ab

where, Δ = √s(s – a)(s – b)(s – c) and s = a + b + c / 2

and A + B + C = 180°

(ii) Solution when two sides and the included angle are given, then

This is called as Napier’s analogy.

(iii) Solution when one side a and two angles A and B are given, then

C = 180° – (A + B)

b = c sin B / sin C and C = sin C / sin A

(iv) Solution when two sides a, b and the opposite ∠A is given, then

sin B = b / a sin A, C = 180° – (A + B), c = a sin C / sin A

Now, different cases arises here.

(a) If A is an acute angle and a < b sin A, then sin B = b / a sin A gives sin B > 1, which is not possible, so no such triangle is possible.

(b) When A is an acute angle and a = b sin A In this case, only one triangle is possible, which is right angled at B.

(c) If A is an acute angle and a > b sin A In this case, there are two values of B given by

sin B = b sin A / a, say B₁ and B₂ such that B₁ + B₂ = 180°,

side c can be calculated from

c = a sin C / sin A

Circles Connected with Triangle

1. Circumcircle

The circle passing through the vertices of the ΔABC is called the circumcircle. Its radius R is called the circumradius.

R = a / 2 sin A = b / 2 sin B = c / 2 sin C = abc / 4 Δ

2.Incircle

The circle touches the three sides of the A triangle internally is called the inscribed or the incircle of the triangle. Its radius r is called inradius of the circle.

3. Escribed Circle

The circle touches BC and the two sides AB and AC produced of ΔABC externally is called the escribed circle opposite A. Its radius is denoted by r_l.

similarly, r₂ and r₃ denote the radii of the escribed circles opposite angles Band C, respectively. Hence, r_l, r₂ and r₃ are called the exradius of &DeltaABC. Here,

4. Orthocentre and Pedal Triangle

The point of intersection of perpendicular drawn from the vertices on the opposite sides of a triangle is called orthocentre.

The ΔDEF formed by joining the feet of the altitudes is called the pedal triangle.

(i) Distance of the orthocentre of the triangle from the angular points are 2R cos A, 2R co~ B, 2R cos C and its distances from the sides are 2 R cos B cos C, 2 R cos C cos A, 2 R cos A cos B.

(ii) The length of medians AD, BE and CF of a Δ ABC are

AD = 1 / 2 √2b² + 2c² – a²

BE 1 / 2 √2c² + 2a² – b²

CF = 1 / 2 √2a² + 2b² – c²

Radii of the Inscribed and Circumscribed Circles of Regular Polygon

(i) R = a / 2 cosec π / n

(ii) r = a / 2 cot π / n, where a is the length of a side of polygon.

(iii) The area of the polygon = n (Area of AABC)

= 1 / 4na² cot (π / n)

= nr² tan π / n = n / 2 R² sin (2π / n)

Some Properties of Triangle

(i) The mid-point of the hypotenuse of a right angled triangle is equidistant from the three vertices of the triangle.

(ii) The mid-point of the hypotenuse of a right angled triangle is the circumcentre of the triangle.

(iii) If any triangle, the exterior angle is equal to the sum of interior opposite angles.

(iv) In a Δ ABC, if DE||AB, then

AB / DE = BC / DC = AC / EC

(v) In any Δ ABC, if AD is the angle bisector of ∠ BAC, then

BD / DC = AB / AC = c / b

(vi) Appollonius Theorem If AD is median of the &dELTA; ABC, then

AB² + AC² = 2 (AD² + BD²)

(vii) m-n Theorem In a ΔABC, D is a A point on the line BC such that BD : DC= m: n and ∠ADC = θ, ∠BAD = α, ∠DAC = β, then

(a) (m + n) cot θ = m cot α – n cot β

(b) (m + n) cot θ = n cot B – m cot C

(viii) In an isosceles triangle, the median is perpendicular to the base.

(ix) ptolemy’s Theorem In a cyclic quadrilateral ABCD

AB x CD + AD x BC = AC x BD

Heights and Distances

Angle of Elevation

If O be the observer’s eye and OX be the p horizontal line through O. If the object P is at higher level than eye, then ∠POX is called the angle of elevation.

Angle of Depression

If the object P is a lower level than O, then ∠POX is called the angle of depression.

Important Results on Height and Distance

(i) a = h (cot α – cot β)

Inverse Function

If y = f(x) and x = g(y) are two functions such that f (g(y)) = y and g (f(y)) = x, then f and y are said to be inverse of each other

i.e., g = f^-1

IF y = f(x), then x = f^-1(y)

Inverse Trigonometric Functions

If y = sin X^-1, then x = sin^-1 y, similarly for other trigonometric functions.

This is called inverse trigonometric function .

Now, y = sin^-1(x), y ∈ [π / 2 , π / 2] and x ∈ [-1,1].

(i) Thus, sin^-1x has infinitely many values for given x ∈ [-1, 1].

(ii) There is only one value among these values which lies in the interval [π / 2 , π / 2]. This value is called the principal value.

Domain and Range of Inverse Trigonometric FunctionsGraphs of Inverse Trigonometric Functions

Properties of Inverse Trigonometric Functions

Property I

Property II

Property III

Property IV

Property V

Property VI

Property VII

Property VIII

Property IX

Property X

Property XI

Property XII

Important Results

where S_k denotes the sum of the product of x₁,x₂,…x_n takes k at a time.

Inverse Hyperbolic Functions

If sinh y = x, then y is called the inverse hyperbolic sine of x and it is written as y = sinh^-1x.

Similarly, cosh^-1 x, tan h^-1 x etc., can be defined,

Domain and Range of Inverse Hyperbolic Functions

Relation between Inverse Circular Functions and Inverse Hyperbolic Functions

(i) sinh^-1x = – i sin^-1(ix)

(ii) cosh^-1x = – i cos^-1 x .

(iii) tanh-^-1x = – i tan^-1(ix)

Important Results

Trigonometric Equation

An equation involving one or more trigonometrical ratios of unknown angle is called a trigonometric equation .

Solution/Roots of a Trigonometric Equation

A value of the unknown angle which satisfies the given equation, is called a solution or root of the equation.

The trigonometric equation may have infinite number of solutions.

(i) Principal Solution – The least value of unknown angle which satisfies the given equation, is called a principal solution of trigonometric equation.

(ii) General Solution – We know that, trigonometric function are periodic and solution of trigonometric equations can be generalised with the help of the periodicity of the trigonometric functions. The solution consisting of all possible solutions of a trigonometric equation is called its general solution.

Important Results

Important Points to be Remembered

(i) While solving an equation, we have to square it, sometimes the resulting roots does not satisfy the original equation.

(ii) Do not cancel common factors involving the unknown angle on LHS and RHS.Because it may be the solution of given equation.

(iii) (a) Equation involving sec θ or tan θ can never be a solution of the form (2n + 1) π / 2.

(b) Equation involving coseca or cote can never be a solution of the form θ = nπ.
For Class 11

Angle

When a ray OA starting from its initial position OA rotates about its end point 0 and takes the final position OB, we say that angle
AOB (written as ∠ AOB) has been formed. The amount of rotation from the initial side to the terminal side is Called the measure of
the angle.

Positive and Negative Angles

An angle formed by a rotating ray is said to be positive or negative depending on whether it moves in an anti-clockwise or a clockwise direction, respectively.

Measurement of Angles

There are three system for measuring angles,

1. Sexagesimal System/Degree Measure (English System)

In this system, a right angle is divided into 90 equal parts, called degrees. The symbol 1° is used to denote one degree. Each degree is divided into 60 equal parts, called minutes and one minute is divided into 60 equal parts, called seconds. Symbols 1′ and 1″ are used to denote one minute and one second, respectively.

i.e., 1 right angle = 90°
1° = 60′
1′ = 60″

2. Centesimal System (French System)

In this system, a right angle is divided into 100 equal parts, called ‘grades’. Each grade is subdivided into 100 min and each minute is divided into 100 s.

i.e., 1 right angle = 100 grades = 100^g
1^g = 100′
1′ = 100″

3. Circular System (Radian System)

In this system, angle is measured in radian.

A radian is the angle subtended at the centre of a circle by an arc, whose length is equal to the radius of the circle.

The number of radians in an angle subtended by an arc of circle at the centre is equal to arc/radius.

Relationships

(i) π radian = 180°
or 1 radian (180°/π)= 57°16’22”
where, π = 22/7 = 3.14159

(ii) 1° = (π/180) rad = 0.01746 rad

(iii) If D is the number of degrees, R is the number of radians and G is the number of grades in an angle θ, then

(iv) θ = l/r where θ = angle subtended by arc of length / at the centre of the circle, r = radius of the circle.

Trigonometric Ratios

Relation between different sides and angles of a right angled triangle are called trigonometric ratios or T-ratios

Trigonometric (or Circular) Functions

Let X’OX and YOY’ be the coordinate axes. Taking 0 as the centre and a unit radius, draw a circle, cutting the coordinate axes at A,B, A’ and B’, as shown in the figure.

Now, the six circular functions may be defined as under
(i) cos θ = x
(ii) sin θ = y
(iii) sec θ = 1/x, x ≠ 0
(iv) cosec θ = 1/y, y ≠ 0
(v) tan θ = y/x, x ≠ 0
(vi) cot θ = x/y, y ≠ 0

Domain and Range

Range of Modulus Functions

|sin θ|≤ 1, |cos θ| ≤ 1, |sec θ| ≥ 1, |Cosec θ| ≥ 1 for all values of 0, for which the functions are defined.

Trigonometric Identities

An equation involving trigonometric functions which is true for all those angles for which the functions are defined is called trigonometrical identity. Some identities are

Sign of Trigonometric Ratios

Trigonometric Ratios of Some Standard Angles

Trigonometric Ratios of Some Special Angles

Trigonometric Ratios of Allied Angles

Two angles are said to be allied when their sum or difference is either zero or a multiple of 90°. The angles — θ, 90° ± θ, 180° ± θ, 270° + θ, 360° —θ etc., are angles allied to the angle θ, if θ is measured in degrees.

Trigonometric Periodic Functions

A function f(x) is said to be periodic, if there exists a real number T> 0 such that f(x + T)= f(x) for all x. T is called the period of the function, all trigonometric functions are periodic.

Maximum and Minimum Values of Trigonometric Expressions

Trigonometric Ratios of Compound Angles

The algebraic sum of two or more angles are generally called compound angles and the angles are known as the constituent angle. Some standard formulas of compound angles have been given below.

Transformation Formulae

Trigonometric Ratios of Multiple Angles

Trigonometric Ratios of Some Useful Angles

Hyperbolic Functions

The hyperbolic functions sinh , cosh z, tanh z, cosech z, sec h z , coth z are angles of the circular functions, defined by removing is appearing in the complex exponentials.

Domain and Range of Hyperbolic Function

identities

(i) cosh²x – sinh²x = 1
(ii) sech²x + tanh²x =1
(iii) coth²x – cosech²x = 1
(iv) cosh²x + sinh²x = cosh²x

Formulae for the Sum and Difference

Formulae to Transform the Product into Sum or Difference

Formulae for Multiples of x

Binomial Theorem for Positive Integer

If n is any positive integer, then

This is called binomial theorem.

Here, ⁿC₀, ⁿC₁, ⁿC₂, … , ⁿn_o are called binomial coefficients and

ⁿC_r = n! / r!(n – r)! for 0 ≤ r ≤ n.

Properties of Binomial Theorem for Positive Integer

(i) Total number of terms in the expansion of (x + a)ⁿ is (n + 1).

(ii) The sum of the indices of x and a in each term is n.

(iii) The above expansion is also true when x and a are complex numbers.

(iv) The coefficient of terms equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficients and

ⁿC_r = ⁿC_{n – r}, r = 0,1,2,…,n.

(v) General term in the expansion of (x + c)ⁿ is given by

T_{r + 1} = ⁿC_rx^{n – r} a^r.

(vi) The values of the binomial coefficients steadily increase to maximum and then steadily decrease .

(vii)

(viii)

(ix) The coefficient of x^r in the expansion of (1+ x)ⁿ is ⁿC_r.

(x)

(xi) (a)

(b)

(xii) (a) If n is odd, then (x + a)ⁿ + (x – a)ⁿ and (x + a)ⁿ – (x – a)ⁿ both have the same number of terms equal to (n +1 / 2).

(b) If n is even, then (x + a)ⁿ + (x – a)ⁿ has (n +1 / 2) terms. and (x + a)ⁿ – (x – a)ⁿ has (n / 2) terms.

(xiii) In the binomial expansion of (x + a)ⁿ, the r th term from the end is (n – r + 2)th term fromthe beginning.

(xiv) If n is a positive integer, then number of terms in (x + y + z)ⁿ is (n + l)(n + 2) / 2.

Middle term in the Expansion of (1 + x)ⁿ

(i) It n is even, then in the expansion of (x + a)ⁿ, the middle term is (n/2 + 1)^th terms.

(ii) If n is odd, then in the expansion of (x + a)ⁿ, the middle terms are (n + 1) / 2 th term and (n + 3) / 2 th term.

Greatest Coefficient

(i) If n is even, then in (x + a)ⁿ, the greatest coefficient is ⁿC_{n / 2}

(ii) Ifn is odd, then in (x + a)ⁿ, the greatest coefficient is ⁿC_{n – 1 / 2} or ⁿC_{n + 1 / 2} both being equal.

Greatest Term

In the expansion of (x + a)ⁿ

(i) If n + 1 / x/a + 1 is an integer = p (say), then greatest term is T_p == T_{p + 1}.

(ii) If n + 1 / x/a + 1 is not an integer with m as integral part of n + 1 / x/a + 1, then T_{m + 1}. is the greatest term.

Important Results on Binomial Coefficients

Divisibility Problems

From the expansion, (1+ x)ⁿ = 1+ ⁿC₁x + ⁿC₁x²+ … +ⁿC_nxⁿ

We can conclude that,

(i) (1+ x)ⁿ – 1 = ⁿC₁x + ⁿC₁x²+ … +ⁿC_nxⁿ is divisible by x i.e., it is multiple of x.

(1+ x)ⁿ – 1 = M(x)

(ii)

(iii)

Multinomial theorem

For any n ∈ N,

(i)

(ii)

(iii) The general term in the above expansion is

(iv)The greatest coefficient in the expansion of (x₁ + x₂ + … + x_m)ⁿ iswhere q and r are the quotient and remainder respectively, when n is divided by m.

(v) Number of non-negative integral solutions of x₁ + x₂ + … + x_n = n is ^{n + r – 1}C_{r – 1}

R-f Factor Relations

Here, we are going to discuss problem involving (√A + B)sup>n = I + f, Where I and n are positive integers.

0 le; f le; 1, |A – B²| = k and |√A – B| < 1

Binomial Theorem for any Index

If n is any rational number, then

(i) If in the above expansion, n is any positive integer, then the series in RHS is finite otherwise infinite.

(ii) General term in the expansion of (1 + x)ⁿ is T_{r + 1 = n(n – 1)(n – 2)… [n – (r – 1)] / r! * x^r}

(iii) Expansion of (x + a)ⁿ for any rational index

(vii) (1 + x)^{– 1} = 1 – x + x² – x³ + …∞

(viii) (1 – x)^{– 1} = 1 + x + x² + x³ + …∞

(ix) (1 + x)^{– 2} = 1 – 2x + 3x² – 4x³ + …∞

(x) (1 – x)^{– 2} = 1 + 2x + 3x² – 4x³ + …∞

(xi) (1 + x)^{– 3} = 1 – 3x + 6x² – …∞

(xii) (1 – x)^{– 3} = 1 + 3x + 6x² – …∞

(xiii) (1 + x)ⁿ = 1 + nx, if x², x³,… are all very small as compared to x.

Important Results

(i) Coefficient of x^m in the expansion of (ax^p + b / x^q)ⁿ is the coefficient of T_{r + l} where r = np – m / p + q

(ii) The term independent of x in the expansion of ax^p + b / x^q)ⁿ is the coefficient of T_{r + l}where r = np / p + q

(iii) If the coefficient of rth, (r + l)th and (r + 2)th term of (1 + x)ⁿ are in AP, then n² – (4r+1) n + 4r² = 2

(iv) In the expansion of (x + a)ⁿ

T_{r + 1} / T_r = n – r + 1 / r * a / x

(v) (a) The coefficient of x^{n – 1} in the expansion of

(x – l)(x – 2) ….(x – n) = – n (n + l) / 2

(b) The coefficient of x^{n – 1} in the expansion of

(x + l)(x + 2) ….(x + n) = n (n + l) / 2

(vi) If the coefficient of pth and qth terms in the expansion of (1 + x)ⁿ are equal, then p + q = n + 2

(vii) If the coefficients of x^r and x^{r + 1} in the expansion of a + x / b)ⁿ are equal, then

n = (r + 1)(ab + 1) – 1

(viii) The number of term in the expansion of (x₁ + x₂ + … + x_r)_{n is n + r – 1C r – 1.}

(ix) If n is a positive integer and a₁, a₂, … , a_m ∈ C, then the coefficient of x^r in the expansion of (a₁ + a₂x + a₃x² +… + a_mx^{m – 1})ⁿ is

(x) For |x| < 1,

(a) 1 + x + x² + x³+ … + ∞ = 1 / 1 – x

(b) 1 + 2x + 3x² + … + ∞ = 1 / (1 – x)²

(xi) Total number of terms in the expansion of (a + b + c + d)ⁿ is (n + l)(n + 2)(n + 3) / 6.

Important Points to be Remembered

(i) If n is a positive integer, then (1 + x)ⁿ contains (n +1) terms i.e., a finite number of terms. When n is general exponent, then the expansion of (1 + x)ⁿ contains infinitely many terms.

(ii) When n is a positive integer, the expansion of (l + x)ⁿ is valid for all values of x. If n is general exponent, the expansion of (i + x)ⁿ is valid for the values of x satisfying the condition |x| < 1.

Fundamental Principles of Counting

1. Multiplication Principle

If first operation can be performed in m ways and then a second operation can be performed in n ways. Then, the two operations taken together can be performed in mn ways. This can be extended to any finite number of operations.

2. Addition Principle

If first operation can be performed in m ways and another operation, which is independent of the first, can be performed in n ways. Then, either of the two operations can be performed in m + n ways. This can be extended to any finite number of exclusive events.

Factorial

For any natural number n, we define factorial as n ! or n = n(n – 1)(n – 2) … 3 x 2 x 1 and 0!= 1!= 1

Permutation

Each of the different arrangement which can be made by taking some or all of a number of things is called a permutation.

Mathematically  The number of ways of arranging n distinct objects in a row taking r (0 ≤ r ≤ n) at a time is denoted by P(n ,r) or ⁿp_r

Properties of Permutation

Important Results on’Permutation

1. The number of permutations of n different things taken r at a time, allowing repetitions is n^r.
2.  The number of permutations of n different things taken all at a time is ⁿP_n= n! .
3. The number of permutations of n things taken all at a time, in which p are alike of one kind, q are alike of second kind and r are alike of third kind and rest are different is n!/(p!q!r!)
4. The number of permutations of n things of which p₁ are alike of one kind p₂ are alike of second kind, p₃ are alike of third kind,…, P_r are alike of rth kind such that p₁ + p₂ + p₃ +…+p_r = n is n!/P₁!P₂!P₃!….P_r!
5. Number of permutations of n different things taken r at a time,
   when a particular thing is to be included in each arrangement is r.^{n – 1}P_{r – 1}.
   when a particular thing is always excluded, then number of arrangements = ^{n – 1}P_r
6. Number of permutations of n different things taken all at a time, when m specified things always come together is m!(n – m + 1)!.
7. Number of permutations of n different things taken all at a time, when m specified things never come together is n! – m! x (n – m + 1)!.

Division into Groups

(i) The number of ways in which (m + n) different things can be divided into two groups which contain m and n things respectively [(m + n)!/m ! n !].

This can be extended to (m + n + p) different things divided into three groups of m, n, p things respectively [(m + n + p)!/m!n! p!].

(ii) The number of ways of dividing 2n different elements into two groups of n objects each is [(2n)!/(n!)²] , when the distinction can be made between the groups, i.e., if the order of group is important. This can be extended to 3n different elements into 3 groups is [(3n)!/((n!)³].

(iii) The number of ways of dividing 2n different elements into two groups of n object when no distinction can be made between the groups i.e., order of the group is not important is

[(2n)!/2!(n!)²].

This can be extended to 3n different elements into 3 groups is

[(3n)!/3!(n!)³].

The number of ways in which mn different things can be divided equally it into m groups, if order of the group is not important is

[(mn)!/(n!)^m m!].

(v) If the order of the group is important, then number of ways of dividing mn different things equally into m distinct groups is mn

[(mn)!/(n!)^m]

(vi) The number of ways of dividing n different things into r groups is

[rⁿ — ^rC₁(r — 1)ⁿ + ^rC₂(r — 2)ⁿ — ^rC₃(r – 3)ⁿ + …].

(vii) The number of ways of dividing n different things into r groups taking into account the order of the groups and also the order of things in each group is

^n+r-1P_n = r(r + l)(r + 2) … (r + n – 1).

(viii) The number of ways of dividing n identical things among r persons such that each gets 1, 2, 3, … or k things is the coefficient of x^{n – r} in the expansion of (1 + x + x² + … + X^k-1)^r.

Circular Permutation

In a circular permutation, firstly we fix the position of one of the objects and then arrange the other objects in all possible ways.

(i) Number of circular permutations at a time is (n -1)!. If clockwise taken as different. of n and different things taken anti-clockwise orders all are
(ii) Number of circular permutations of n different things taken all at a time, when clockwise or anti-clockwise order is not different 1/2(n – 1)!.
(iii) Number of circular permutations of n different things taken r at a time, when clockwise or anti-clockwise orders are take as different is

ⁿP_r/r.

(iv) Number of circular permutations of n different things taken r at a time, when clockwise or anti-clockwise orders are not different is

ⁿP_r/2r.

(v) If we mark numbers 1 to n on chairs in a round table, then n persons sitting around table is n!.

Combination

Each of the different groups or selections which can be made by some or all of a number of given things without reference to the order of the
things in each group is called a combination.

Mathematically The number of combinations of n different things taken r at a time is

Properties of Combination

Important Results on Combination

• The number of combinations of n different things taken r at a time allowing repetitions is ^{n + r – 1}C_r
• The number of ways of dividing n identical things among r persons such that each one gets at least one is ^{n – 1}C_{r – 1}.
• The total number of combinations of n different objects taken r at a time in which
  (a) m particular objects are excluded = ^{n – m}C_r
  (b) m particular objects are included = ^{n – m}C_{r – 1}
• The total number of ways of dividing n identical items among r persons, each one of whom can receive 0, 1, 2 or more items (≤ n) is ^{n + r – 1}C_{r – 1}
• The number of ways in which n identical items can be divided into r groups so that no group contains less than in items and more than k(m < k) is coefficient of xⁿ in the expansion of (x^m + x^{m + 1} +….+ x^k)^r.
• The total number of ways of selection of some or all of n things at a time is ⁿC₁ + ⁿC₂ +….+ ⁿn₁ = 2ⁿ — 1.
• The number of selections of r objects out of n identical objects is 1.
• Total number of selections of zero or more objects from n identical objects is n + 1.

Important Points to be Remembered

1. Function

(i) If a set A has m elements and set B has n elements, then

(a) number of functions from A to B is n^m
(b) number of one-one function from A to B is ⁿP_m, m ≤ n.
(c) number of onto functions from A to B is n^m — ⁿC₁(n — 1)^m + ⁿC₂(n — 2)^m…..; m ≤ n.
(d) number of increasing (decreasing) functions from A to B is ⁿC_m, m ≤ n.
(e) number of non-increasing (non-decreasing) functions from A to B is ^{m + n – 1}C_m .
(f) number of bijective (one-one onto) functions from A to B is n !, if m = n.

(ii) Number of permutations of n different objects taken r at a time in which m particular objects are always

(a) excluded = ^{n – m}P_r
(b) included = ^{n – m}P_{r – m} x r!

2. Geometry

1. Given, n distinct points in the plane, no three of which are collinear, then the number of line segments formed = ⁿC₂.
2. Given. ii distinct paints in the p)ane. in which m are collinear (m ≥ 3), then the number of line segments is (ⁿC₂ – ^mC₂) + 1.
3. Given, n distinct points in the plane, no three of which are collinear, then the number of triangle formed = ⁿC₃
4. Given, n distinct points in a plane, in which m are collinear (m ≥ 3), then the number of triangle formed = ⁿC₃ — ^mC₃
5. The number of diagonals in a n-sided closed polygon = ⁿC₂ — n.
6. Given, n points on the circumference of a circle, then
   (a) number of straight lines = ⁿC₂
   (b) number of triangles = ⁿC₃
   (c) number of quadrilaterals = ⁿC₄
7. Number of rectangles of any size in a square of n x n isand number of square of any size is.
8. In a rectangle of n x p (n < p), numbers of rectangles of any size is np/4 (n + 1) (p + 1) and number of squares of any size is
   
9. Suppose n straight lines are drawn in the plane such that no two lines are parallel and no three lines are concurrent, then number of parts which these divides the plane is equal to 1 +∑ n.

3. Prime Factors

Any natural number > 1, can be expressed as product of primes.

• Let n = p₁^α1 p₂^α2 p₃^α3 …. p_r^αr, where
• p_i, i = 1, 2, 3, … , r, are prime numbers.
• α_i, i = 1, 2, 3, … , r, are positive integers.

1. Number of distinct positive integral divisors of n is (α₁ + 1)(α₂ + 1)(α₃ + 1) … (α_r + 1).
2. Sum of distinct positive integral divisors of n is
   
3. Total number of divisors of n (excluding 1 and n), is (α₁ + 1)(α₂ + 1)(α₃ + 1) … (α_r + 1) – 2.
4. Total number of divisors of n (excluding 1 or n), is (α₁ + 1)(α₂ + 1)(α₃ + 1) … (α_r + 1) – 1.
5. The number of ways in which n can be resolved as a product of two factors is
   (a) 1/2(α₁ + 1)(α₂ + 1)(α₃ + 1) … (α_r + 1) if n is not a perfect square.
   (b) 1/2[(α₁ + 1)(α₂ + 1)(α₃ + 1) … (α_r + 1) + 1], if n is a perfect square.
6. The number of ways in which n can be resolved into two factors which are prime to each other is 2^{r – 1}, where r is the number of different factors in n.
7. If p is prime and p^r divides n!, then
   

4. Integral Solutions

1. The number of integral solutions of x₁+ x₂ +….+ x_r = n, where x₁, x₂, … x_r ≥ 0 is ^{n + r – 1}C_{r – 1}.
2. Number of integral solutions of x₁+ x₂ +….+ x_r = n, where x₁, x₂, … x_r ≥ 1 is ^{n – 1}C_{r – 1}

5. Sum of Digits

1. Sum of the numbers formed by taking all the given n digits = (Sum of all the n digits) x (n — 1)! x (111… 1)n times.
2. The sum of all digits in the unit place of all numbers formed with the help of ai , a2, , an all at a time is (n — 1)!(a₁ + a₁ + …. + a_n).
3. The sum of all digits of numbers that can be formed by using the digits a₁, a₂,… , a_n(repetition of digits is not allowed (n — 1)! (a₁ + a₁ + … + a_n)((10ⁿ – 1)/9)

6. Arrangements

1. The number of ways in which m (one type of different things) and n (another type of different things) can be arranged in a row so that all the second type of things come together is n !(m + 1)!.
2. The number of ways in which m (one type of different things) and n (another type of different things) can be arranged in row so that no two things of the same type come together is 2 x m! n!
3. The number of ways in which m (one type of different things) and n (another type of different things) (m ≥ n), can be arranged in a circle so that no two things of second type come together (m – 1)! ^mP_n and when things of second type come together = m! n!
4. The number of ways in which m things of one type and n things of another type (all different) can be arranged in the form of a garland so that all the second type of things come together, is m! n!/2 and if no things of second type come together is, [((m – 1)!^mP_n)/2]

7. Dearrangements

If n distinct objects are arranged in a row, then the number of ways in which they can be rearranged so that no one of them occupies the place assigned to it is

8. Selection

1. The total number of ways in which it is possible to make a selection by taking some or all the given n different objects is
   ⁿC₁ + ⁿC₂ + …. + ⁿC_n = 2ⁿ – 1
2. If there are m items of one kind, n items of another kind and so on. Then, the number of ways of choosing r items out of these items = coefficient of x^r in
   (1 + x + x² + …. + x^m)(1 + x + x² + …. + xⁿ)
3. If there are m items of one kind, n items of another kind and so on. Then, the number of ways of choosing r items out of these items such that at least one item of each kind is included in every selection = coefficient of x^r in
   (x + x² + …. + x^m)(x + x² + …. + xⁿ)….
4. The number of ways of selecting r items from a group of n items in which p are identical, is
   ^{n – P}C_r + ^{n – P}C_{r – 1}+ ^{n – P}C_{r – 2} + … + ^{n – P}C₀, if r ≤ P and ^{n – P}C_r + ^{n – P}C_{r – 1}+ ^{n – P}C_{r – 2} + … + ^{n – P}C_{r – p}, if r > P
5. The number of ways in which n identical things can be distributed into r different groups is ^{n + r – l}C_{r – 1}, or ^{n – 1}C_{r – 1} according as blanks groups are or are not admissible.
6. The number of ways of answering one or more of n questions is 2ⁿ – 1.
7. The number of ways of answering one or more n questions when each question has an alternative = 2ⁿ
8. n! + 1 is not divisible by any natural number between 2 and n.
9. If there are 1 objects of one kind, m objects of second kind, n objects of third kind and so on. Then, the number of possible arrangements of r objects out of these objects = Coefficient of x^r in the expansion of

1. Sequence: Sequence is a function whose domain is a subset of natural numbers. It represents the images of 1, 2, 3,… ,n, as f₁, f₂, f₃, …., f_n , where f_n = f(n).

2. Real Sequence: A sequence whose range is a subset of R is called a real sequence.

3. Series: If a₁, a₂, a₃ , … , a_n is a sequence, then the expression a₁ + a₂ + a₃ + … + a_n is a series.

4. Progression: A sequence whose terms follow certain rule is called a progression.

5. Finite Series: A series having finite number of terms is called finite series.

6. Infinite Series: A series having infinite number of terms is called infinite series.

Arithmetic Progression (AP)

A sequence in which the difference of two consecutive terms is constant, is called Arithmetic Progression (AP).

Properties of Arithmetic Progression

(i) If a sequence is an AP, then its nth term is a linear expression in n, i.e., its nth term is given by An + B, where A and B are constants and A = common difference.

(ii) nth Term of an AP If a is the first term, d is the common difference and / is the last term of an AP, then

(a) nth term is given by 1= a_n = a + (n – 1)d

(b) nth term of an AP from the last term is a’_n = l – (n – 1)d

(c) a_n + a’_n = a + 1
i.e., nth term from the start + nth term from the end
= constant
= first term + last term

(d) Common difference of an AP
d = T_n – T_n-1, ∀ n > 1

(e) T_n = 1/2[T_n-k + T_n+k], k < n

(iii) If a constant is added or subtracted from each term of an AP, then the resulting sequence is an AP with same common difference.

(iv) If each term of an AP is multiplied or divided by a non-zero constant k, then the resulting sequence is also an AP, with common difference kd or d/k where d = common difference.

(v) If a_n, a_n+1 and a_n+2 are three consecutive terms of an AP, then 2a_n+1 = a_n + a_n+2.

(vi) (a) Any three terms of an AP can be taken as a – d, a, a + d.

(b) Any four terms of an AP can be taken as a-3d,a- d, a + d, a + 3d.

(c) Any five terms of an AP can be taken as a-2d,a – d, a, a + d, a + 2d.

(vii) Sum of n Terms of an AP

(a) Sum of n terms of AP, is given by S_n = n/2[2a + (n – 1)d] = n/2[a + l]

(b) A sequence is an AP, iff the sum of n terms is of the form An² + Bn, where A and B are constants. Common difference in such case will be 2A.

(c) T_n = S_n – S_n-1

(viii) a², b² and c² are in AP.

(ix) If a₁, a₂,…, a_n are the non-zero terms of an AP, then

(x) Arithmetic Mean

(a) If a, A and b are in AP, then A= (a + b)/2 is called the 2 arithmetic mean of a and b.

(b) If a₁, a₂, a₃ , an are n numbers, then their AM is given by,

(c) If a, A₁ , A₂ , A₃ ,…,A_n, b are in AP, then A₁, A₂, A₃,…, A_n are n arithmetic mean between a and b, where

(d) Sum of n AM’s between a and b is nA
i.e., A₁ + A₂ + A₃ + + = nA

Geometric Progression (GP)

A sequence in which the ratio of two consecutive terms is constant is called GP. The constant ratio is called common ratio (r).
i.e., a_n+1/a_n = r, ∀ n ≥ 1

Properties of Geometric Progression (GP)

(i) nth Term of a GP If a is the first term and r is the common ratio

(a) nth term of a GP from the beginning is a_n = ar^n-1
(b) nth term of a GP from the end is a’_n = l/r^n-1, l = last term
(c) If a is the first term and r is the common ratio of a GP, then the GP can be written as a, ar, ar²,… , ar^n-1, …
(d) The nth term from the end of a finite GP consisting of m terms is ar^m-n, where a is the first term and r is the common ratio of the GP.
(e) a_na’_n = al i.e., nth term from the beginning x nth term from the end = constant = first term x last term.

(ii) If all the terms of GP be multiplied or divided by same non-zero constant, then the resulting sequence is a GP with the same common ratio.

(iii) The reciprocal terms of a given GP form a GP.

(iv) If each term of a GP be raised to same power, the resulting sequence also forms a GP.

(v) If the terms of a GP are chosen at regular intervals, then the resulting sequence is also a GP.

(vi) If a₁, a₂, a₃, … , a_n are non-zero, non-negative term of a GP, then

(a) GM = (a₁a₂a₃… a_n )^1/n
(b) log a₁, log a₂, log a₃,…, log a_n are in an AP and vice-versa.

(vii) If a, b and c are three consecutive terms of a GP, then b² = ac

(viii) (a) Three terms of a GP can be taken as a/r, a and ar.
(b) Four terms of a GP can be taken as a/r³, a/r, ar and ar³.
(c) Five terms of a GP can be taken as a/r², a/r, ar and ar².

(ix) Sum of n Terms of a GP

(a) Sum of n terms of a GP is given by

(x) Geometric Mean (GM)

(a) If a, G, b are in GP, then G is called the geometric mean of a and b and is given by G = √ab
(b) If a, G₁, G₂, G₃, , G_n, b are in GP, then G₁, G₂, G₃,… , G_n, are in GM’s between a and b, where

(c) Product of n GM’s, G₁ X G₂ X G₃ X … X G_n = Gⁿ

Harmonic Progression (HP)

A sequence a₁, a₂, a₃ ,…, a_n of non-zero numbers is called a Harmonic Progression (HP), if the sequence 1/a₁, 1/a₂, 1/a₃, …, 1/a_n is an AP.

Properties of Harmonic Progression (HP)

(i) nth term of HP, if a₁, a₂, a₃ ,…, a_n are in HP, then
(a) nth term of the HP from the beginning

(b) nth term of the HP from the end

(d) a_n = 1/a+(n-1)d are the first term and common difference of the corresponding AP.
(ii) Sum of harmonic progression does not exist.

Harmonic Mean

(i) If a, H,b are in HP, then H is called the harmonic mean of a and b i.e., H= 2ab/(a + b)

(ii) If a, H₁, H₂, H₃, …, H_n, b are in HP, then
H₁, H₂, H₃, …, H_n

are n harmonic means between a and b where

(iii) Harmonic Mean (HM) between a₁, a₂, a₃, …, a_n is given by

Properties of AM, GM and HM between Two Numbers

If A, G and H are arithmetic, geometric and harmonic means of two positive numbers a and b, then
(i) A=(a+b)/2, G=√ab, H=(2ab)/(a+b)
(ii) A≥G≥H
(iii) A, G, H are in GP and G² = AH
(iv) If A,G,H are AM, GM and HM between three given numbers a, b and c, then the equation on having a, b and c as its root is

(v) If A1,A2 be two AM’s, G₁, G₂ be two GM’s and H₁, H₂ be two HM’s between two numbers a and b, then

(vi) If A,G and H be AM, GM and HM between two numbers a and b, then

Arithmetico-Geometric Progression

A sequence in which every term is a product of a term of AP and GP is known as arithmetico-geometric progression.

The series may be written as

Sum of Arithmetico-Geometric Series

Type 1 Let a_l + a₂ + a₃ + … be a given series. If a₂ – a_l, a₃ – a₂, … are in AP or GP, then a_n and S_n can be found by the method of difference.

where T₁, T₂, T₃ ,… are terms of new series and S_n = Σa_n

Type 2 It is not always necessary that the series of first order of differences i.e., a₂ – a₁, a₃ – a₂, …, a_n – a_n-1 is always either in AP or in GP in such case.

Now, the series (T₂ — T₁) + (T₃ — T₂) + …+ (T_n – T_n-1) is series of second order of differences and when it is either in AP or in GP, then a_n = a₁ + ΣT_r

Otherwise, in the similar way, we find series of higher order of differences and the nth term of the series.

Exponential Series

The sum of the series is denoted by thenumber e.

(i) e lies between 2 and 3.
(ii) e is an irrational number.

Exponential Theorem

Let a>0, then for all real value of x,

Logarithmic Series

Important Result and Useful Series

24. If number of terms in AP/GP/HP are odd, then AM/GM/HM of first and last term in middle term of progression.

25. If pth, qth and rth term of geometric progression are also in geometric progression.

26. If a,b and c are in AP and also in GP, then a=b=c

27. If a, b and c are in AP, then xa, xb and xc are in geometric progression.

1. Real Polynomial: Let a₀, a₁, a₂, … , a_n be real numbers and x is a real variable. Then, f(x) = a₀ + a₁x + a₂x² + … + a_nxⁿ is called a real polynomial of real variable x with real coefficients.

2. Complex Polynomial: If a₀, a₁, a₂, … , a_n be complex numbers and x is a varying complex number, then f(x) = a₀ + a₁x + a₂x² + … + a_{n – 1}x^{n – 1} + a_nxⁿ is called a complex polynomial or a polynomial of complex variable with complex coefficients.

3. Degree of a Polynomial: A polynomial f(x) = a₀ + a₁x + a₂x² + a₃x³ + … + a_nxⁿ , real or complex is a polynomial of degree n , if a_n ≠ 0.

4. Polynomial Equation: If f(x) is a polynomial, real or complex, then f(x) = 0 is called a polynomial equation. If f(x) is a polynomial of second degree, then f(x) = 0 is called a quadratic equation .

Quadratic Equation: A polynomial of second degree is called a quadratic polynomial. Polynomials of degree three and four are known as cubic and biquadratic polynomials respectively. A quadratic polynomial f(x) when equated to zero is called quadratic equation. i.e., ax² + bx + c = 0 where a ≠ 0.

Roots of a Quadratic Equation: The values of variable x .which satisfy the quadratic equation is called roots of quadratic equation.

Important Points to be Remembered

• An equation of degree n has n roots, real or imaginary .
• Surd and imaginary roots always occur in pairs of a polynomial equation with real coefficients i.e., if (√2 + √3i) is a root of an equation, then’ (√2 – √3i) is also its root. .
• An odd degree equation has at least one real root whose sign is opposite to that of its last’ term (constant term), provided that the coefficient of highest degree term is positive.
• Every equation of an even degree whose constant term is negative and the coefficient of highest degree term is positive has at least two real roots, one positive and one negative.
• If an equation has only one change of sign it has one positive root.
• If all the terms of an equation are positive and the equation involves odd powers of x, then all its roots are complex.

Solution of Quadratic Equation

1.Factorization Method: Let ax² + bx + c = α(x – α) (x – β) = O. Then, x = α and x = β will satisfy the given equation.

2. Direct Formula: Quadratic equation ax² + bx + c = 0 (a ≠ 0) has two roots, given by

where D = Δ = b² – 4ac is called discriminant of the equation .

Above formulas also known as Sridharacharya formula.

Nature of Roots

Let quadratic equation be ax² + bx + c = 0, whose discriminant is D.

(i) For ax² + bx + c = 0; a, b , C ∈ R and a ≠ 0, if

(a) D < => Complex roots

(b) D > 0 => Real and distinct roots

(c) D = 0 => Real and equal roots as α = β = – b/2a

(ii) If a, b, C ∈ Q, a ≠ 0, then

(a) If D > 0 and D is a perfect square => Roots are unequal and rational.

(b) If D > 0, a = 1; b, c ∈ I and D is a perfect square. => Roots are integral. .

(c) If D > and D is not a perfect square. => Roots are irrational and unequal.

(iii) Conjugate Roots The irrational and complex roots of a quadratic equation always occur in pairs. Therefore,

(a) If one root be α + iβ, then other root will be α – iβ.

(b) If one root be α + √β, then other root will be α – √β.

(iv) If D, and D2 be the discriminants of two quadratic equations, then

(a) If D₁ + D₂ ≥ 0, then At least one of D₁ and D₂ ≥ 0 If D₁ < 0, then D₂ > 0 ,

(b) If D₁ + D₂ < 0, then At least one of D₁ and D₂ < 0 If D₁ > 0, then D₂ < 0

Roots Under Particular Conditions

For the quadratic equation ax² + bx + e = 0.

(i) If b = 0 => Roots are real/complex as (c < 0/c > 0) and equal in magnitude but of opposite sign.

(ii) If c = 0 => One roots is zero, other is – b / a.

(iii) If b = C = 0 => Both roots are zero.

(iv) If a = c => Roots are reciprocal to each other.

(v) If a > 0, c < 0, a < 0, c > 0} => Roots are of opposite sign.

(vi) If a > 0, b > 0, c > 0, a < 0, b < 0, c < 0} => Both roots are negative, provided D ≥ 0

(vii) If a > 0, b < 0, c > 0, a < 0, b > 0, c < 0} => Both roots are positive, provided D ≥ 0

(viii) If sign of a = sign of b ≠ sign of c => Greater root in magnitude is negative.

(ix) If sign of b = sign of c ≠. sign of a => Greater root in magnitude is positive.

(x) If a + b + c = 0 => One root is 1 and second root is c/a.

Relation between Roots and Coefficients

1. Quadratic Equation: If roots of quadratic equation ax² + bx + c = 0 (a ≠ 0) are α and β, then Sum of roost = S = α + β = -b/a = – coefficient of x / coefficient of x² Product of roots = P = α * β = c/a = constant term / coefficient of x²

2. Cubic Equation: If α, β and γ are the roots of cubic equation ax³ + bx² + cx + d = 0.

Then,

3. Biquadratic Equation: If α, β, γ and δ are the roots of the biquadratic equation ax⁴ + bx³ + cx² + dx + e = 0, then

Symmetric Roots: If roots of quadratic equation ax² + bx + c = 0 (a ≠ 0) are α and β, then

Formation of Polynomial Equation from Given Roots

If a₁, a₂ a₃,…, an are the roots of an nth degree equation, then the equation is xⁿ – S₁X^{n – 1}+ S₂X^{n – 2} – S₃X^{n – 3} +…+( _l)ⁿ S_n = 0 where S_n denotes the sum of the products of roots taken n at a time.

1. Quadratic Equation

If α and β are the roots of ‘a quadratic equation, then the equation is x² – S₁X + S₂ = 0

i.e., x² – (α + β) x + αβ = 0

2. Cubic Equation

If α, β and γ are the roots of cubic equation, then the equation is

3. Biquadratic Equation

If α, β, γ and δ are the roots of a biquadratic equation, then the equation is

Equation In Terms of the Roots of another Equation

If α, β are roots of the equation ax² + bx + c = 0, then the equation whose roots are.

The quadratic function f(x) = ax² + 2hxy + by² + 2gx + 2fy + c is always resolvable into linear factor, iff

abc + 2fgh – af² – bg² – ch² = 0

Condition for Common Roots in a Quadratic Equation

1. Only One Root is Common

If α be the common root of quadratic equations

a₁x² + b₁x + C₁ = 0,

and a₂x² + b₂x + C₂ = 0,

then a₁a² + b₁α + C₁ = 0,

and a₂a² + b₂α + C₂ = 0,

By Cramer’s Rule

Hence, the condition for only one root common is

(c₁a₂ – c₂a₁)₂ = (b₁c₂ – b₂c₁)(a₁b₂ – a₂b₁)

2. Both Roots are Common

The required condition is

a₁ / a₂ = b₁ / b₂ = c₁ / c₂

(i) To find the common root of two equations, make the coefficient of second degree term in the two equations equal and subtract. The value of x obtained is the required common root.

(ii) Two different quadratic equations with rational coefficient can not have single common root which is complex or irrational as imaginary and surd roots always occur in pair.

Properties of Quadratic Equation

(i) f(a) . f(b) < 0, then at least one or in general odd number of roots of the equation f(x) = 0 lies between a and b.

(ii) f( a) . f( b) > 0, then in general even number of roots of the equation f(x) = 0 lies between a and b or no root exist f(a) = f(b), then there exists a point c between a and b such that f'(c) = 0, a < c < b.

(iii) If the roots of the quadratic equation a₁x² + b₁x + c₁ = 0, a₂x² + b₂x + c₂ = 0 are in the ratio (i.e., &alpha₁;/β₁ = &alpha₂;/β₂), then

b₁² / b₂² = a₁c₁ / a₂c₂.

(iv) If one root is k times the other root of the quadratic equation ax² + bx + c = 0 ,then

(k + 1)² / k = b² / ac

Quadratic Expression

An expression of the form ax² + bx + c, where a, b, c ∈ R and a ≠ 0 is called a quadratic expression in x .

1. Graph of a Quadratic Expression

We have

y = ax² + bx + c = f(x)

Let y + D/4a = Y and x + D / 2a = X

Y = a * X² => X² = Y / a

(i) The graph of the curve y = f(x) is parabolic.
(ii) The axis of parabola is X = 0 or x + b / 2a = 0 i.e., (parallel to Y-axis). •
(iii) If a > 0, then the parabola opens upward.
If a < 0, then the parabola opens downward.

2. Position of y = ax² +bx + c with Respect to Axes.

(i) For D > 0, parabola cuts X-axis in two real and distinct points
i.e, x = -b ± √D / 2a

(ii) For D = 0, parabola touch X-axis in one point, x = – b/2a.

(iii) For D < O,parabola does not cut X-axis (i.e., imaginary value of x).

3. Maximum and Minimum Values of Quadratic Expression

(i) If a > 0, quadratic expression has least value at x = b / 2a. This least value is given by 4ac – b² / 4a = – D/4a. But their is no greatest
value.

(ii) If a < 0, quadratic expression has greatest value at x = – b/2a . This greatest value is given by 4ac – b² / 4a = – D/4a. But their is no least value.

4. Sign of Quadratic Expression

(i) a > 0 and D < 0, so f(x) > 0 for all x ∈ R i.e., f(x) is positive for all real values of x.

(ii) a < 0 and D < 0, so f(x) < 0 for all x ∈ R i.e., f(x) is negative for all real values of x.

(iii) a > 0 and D = 0, so f(x) ≥ 0 for all x ∈ R i.e., f(x) is positive for all real values of x except at vertex, where f(x) = 0.

(iv) a < 0 and D = 0, so f(x) ≤ 0 for all x ∈ R i.e., f(x) is negative for all real values of x except at vertex, where f(x) = 0.

(v) a > 0 and D > 0
Let f(x) = o have two real roots α and β (α < β), then f(x) > 0 for x ∈ (- ∞, α) ∪ (β,∞) and f (x) < 0 for all x ∈ (α, β).

(vi) a < 0 and D > 0
Let f(x) = 0 have two real roots α and β (α < β). Then, f(x) < 0 for all x ∈ (- ∞, α) ∪ (β,∞) and f(x) > 0 for all
x ∈ (α, β). ,

5. Intervals of Roots

In some problems, we want the roots of the equation ax² + bx + c = 0 to lie in a given interval. For this we impose conditions on a, b and c.
Since, a ≠ 0, we can take f(x) = x² + b/a x + c/a.

(i) Both the roots are positive i.e., they lie in (0,∞), if and only if roots are real, the sum of the roots as well as the product of the roots is positive.

α + β = -b/a > 0 and αβ = c/a > 0 with b² – 4ac ≥ 0

Similarly, both the roots are negative i.e., they lie in (- ∞,0) ifF roots are real, the sum of the roots is negative and the product of the roots is positive.

i.e., α + β = -b/a < 0 and αβ = c/a > 0 with b² – 4ac ≥ 0

(ii) Both the roots are greater than a given number k, iFf the following conditions are satisfied

D ≥ 0, -b/2a > k and f(k) > 0

(iii) Both .the roots are less than a given number k, iff the following conditions are satisfied

D ≥ 0, -b/2a > k and f(k) > 0

(iv) Both the roots lie in a’ given interval (k₁, k₂), iff the following conditions are satisfied

D ≥ 0,k₁ < -b/2a < k₂ and f(k₁) > 0, f(k₂) > 0

(v) Exactly one of the roots lie in a given interval (k₁, k₂), iff

f(k₁) f(k₂) < 0

(vi) A given number k lies between the roots iff f(k) < O. In particular, the roots of the equation will be of opposite sign, iff 0 lies between the roots.

⇒ f(0) < 0

Wavy Curve Method

Let f(x) = (x – a₁)^k₁ (x – a₂)^k₂(x — a₃)^k₃ … (x – a_{n – 1})^k_{n – 1} (x – a_n)^k_n

where k₁, k₂, k₃,…, k_n ∈ N and a₁, a₂, a₃,…, a_n are fixed natural numbers satisfying the condition.

a₁ < a₂ < a₃ < … < a_{n – 1} < a_n..

First we mark the numbers a₁, a₂, a₃,…, a_n on the real axis and the plus sign in the interval of the right of the largest of these numbers, i.e., on the right of a_n. If k_n is even, we put plus sign on the left of a_n and if k_n is odd, then we put minus sign on the left of a_n In the next interval we put a sign according to the following rule.

When passing through the point a_{n – 1} the polynomial f(x) changes sign . if k_{n – 1} is an odd number and the polynomial f(x) has same sign if k_{n – 1} is an even number. Then, we consider the next interval and put a sign in it using the same rule.

Thus, we consider all the intervals. The solution of f(x) > 0 is the union of all interval in which we have put the plus sign and the solution of f(x) < 0 is the union of all intervals in which we have put the minus Sign.

Descarte’s Rule of Signs

The maximum number of positive real roots of a polynomial equation f(x) = 0 is the number of changes of sign from positive to negative and negative to positive in f(x) .

The maximum number of negative real roots of a polynomial equation f(x) = 0 is the number of changes of sign from positive to negative and negative to positive in f(x).

Rational Algebraic In equations

(i) Values of Rational Expression P(x)/Q(x) for Real Values of x, where P(x) and Q(x) are Quadratic Expressions To find the values attained by rational expression of the form a₁x² + b₁x + c₁ / a₂x² + b₂x + c₂

for real values of x.

(a) Equate the given rational expression to y.
(b) Obtain a quadratic equation in x by simplifying the expression,
(c) Obtain the discriminant of the quadratic equation.
(d) Put discriminant ≥ 0 and solve the in equation for y. The values of y so obtained determines the set of values attained by the given rational expression.

(ii) Solution of Rational Algebraic In equation If P(x) and Q(x) are polynomial in x, then the in equation P(x) / Q(x) > 0,
P(x) / Q(x) < 0, P(x) / Q(x) ≥ 0 and P(x) / Q(x) ≤ 0 are known as rational algebraic in equations.

To solve these in equations we use the sign method as

(a) Obtain P(x) and Q(x).
(b) Factorize P(x) and Q(x) into linear factors.
(c) Make the coefficient of x positive in all factors.
(d) Obtain critical points by equating all factors to zero.
(e) Plot the critical points on the number line. If these are n critical points, they divide the number line into (n + 1) regions.
(f) In the right most region the expression P(x) / Q(x) bears positive sign and in other region the expression bears positive and negative signs depending on the exponents of the factors .

Lagrange’s identity

If a₁, a₂, a₃, b₁, b₂, b₃ ≠ R, then

(a₁² + a₂² + a₃²) (b₁² + b₂² + b₃²) – (a₁b₁ + a₂b₂ + a₃b₃)²

= (a₁b₂ – a₂b₁)² + (a₂b₃ – a₃b₂ )² + (a₃b₁ – a₁b₃)²

Algebraic Interpretation of Rolle’s Theorem

Let f (x) be a polynomial having α and β as its roots such that α < β, f(α) = f(β) = 0.Also, a polynomial function is everywhere continuous and differentiable, then there exist θ ∈ (α, β) such that f'(θ) = 0. Algebraically, we can say between any two zeros of a polynomial f(x) there is always a derivative f’ (x) = 0.

Equation and In equation Containing Absolute Value

1. Equation Containing Absolute Value

By definition, |x| = x, if x ≥ 0 OR -x, if x < 0

If |f(x) + g(x)| = |f(x)| + g(x)|, then it is equivalent to the system f(x) . g(x) ≥ 0.

If |f(x) – g(x)| = |f(x)| – g(x)|, then it is equivalent to the system f(x) . g(x) ≤ 0.

2.In equation Containing Absolute Value

(i) |x| < a ⇒ – a < x < a (a > 0)
(ii) |x| ≤ a ⇒ – a ≤ x ≤ a
(iii) |x| > a ⇒ x < – a or x > a
(iv) |x| ≥ a ⇒ x le; – a or x ≥ a

3. Absolute Value of Real Number

|x| = -x, x < 0 OR +x, x ≥ 0

(i) |xy| = |x||y|
(ii) |x / y| = |x| / |y|
(iii) |x|² = x²
(iv) |x| ≥ x
(v) |x + y| ≤ |x| + |y|
Equality hold when x and y same sign.
(vi) |x – y| ≥ ||x| – |y||

Inequalities

Let a and b be real numbers. If a – b is negative, we say that a is less than b (a < b) and if a – b is positive, then a is greater than b (a > b).

Important Points to be Remembered

(i) If a > b and b > c, then a > c. Generally, if a₁ > a₂, a₂ > a₃,…., a_{n – 1} > a_n, then a₁ > a_n.

(vii) If a < x < b and a, b are positive real numbers then a² < x² < b²

Important Inequality

1. Arithmetico-Geometric and Harmonic Mean Inequality

(i) If a, b > 0 and a ≠ b, then

(ii) if a_i > 0, where i = 1,2,3,…,n, then

(iii) If a₁, a₂,…, a_n are n positive real numbers and m₁, m₂,…,m_n are n positive rational numbers, then

i.e., Weighted AM > Weighted GM

(iv) If a₁, a₂,…, a_n are n positive distinct real numbers, then

(a)

(b)

(c) If a₁, a₂,…, a_n and b₁, b₂,…, b_n are rational numbers and M is a rational number, then

(d)

(v) If a₁, a₂, a₃,…, a_n are distinct positive real numbers and p, ,q, r are natural numbers, then

2. Cauchy – Schwartz’s inequality

If a₁, a₂,…, a_n and b₁, b₂,…, b_n are real numbers, such that

(a₁b₁ + a₂b₂ + …+ a_nb_n)² ≤ (a₁² + a₂² + …, a_n²) * (b₁² + b₂² + …, b_n²)

Equality holds, iff a₁ / b₁ = a₂ / b₂ = a_n / b_n

3. Tchebychef’s Inequality

Let a₁, a₂,…, a_n and b₁, b₂,…, b_n are real numbers, such that

(i) If a₁ ≤ a₂ ≤ a₃ ≤… ≤ a_n and b₁ ≤ b₂ ≤ b₃ ≤… ≤ b_n, then

n(a₁b₁ + a₂b₂ + a₃b₃ + …+ a_nb_n) ≥ (a₁ + a₂ + …+ a_n) (b₁ + b₂ + …+ b_n)

(ii) If If a₁ ≥ a₂ ≥ a₃ ≥… ≥ a_n and b₁ ≥ b₂ ≥ b₃ ≥… ≥ b_n, then

n(a₁b₁ + a₂b₂ + a₃b₃ + …+ a_nb_n) ≤ (a₁ + a₂ + …+ a_n) (b₁ + b₂ + …+ b_n)

4. Weierstrass Inequality

(i) If a₁, a₂,…, a_n are real positive numbers, then for n ≥ 2

(1 + a₁)(1 + a₂) … (1 + a_n) > 1 + a₁ + a₂ + … + a_n

(ii) If a₁, a₂,…, a_n are real positive numbers, then

(1 – a₁)(1 – a₂) … (1 – a_n) > 1 – a₁ – a₂ – … – a_n

5. Logarithm Inequality

(i) (a) When y > 1 and log_y x > z ⇒ x > y^z

(b) When y > 1 and log_y x < z ⇒ 0 < x < y^z

(ii) (a) When 0 < y < 1 and log_y x > z ⇒ 0 < x < y^z

(b) hen 0 < y < 1 and log_y x < z ⇒ x > y^z

Application of Inequalities to Find the Greatest and Least Values

(i) If x_l,x₂,…,x_n are n positive variables such that x_l + x₂ +…+ x_n = c (constant), then the product x_l * x₂ *….* x_n is greatest when x₁ = x₂ =… =x_n = c/n and the greatest value is (c/n)ⁿ.

(ii) If x_l,x₂,…,x_n are positive variables such that x_l,x₂,…,x_n = c (constant), then the sum x_l+ x₂ +….+ x_n is least when x₁ = x₂ =… =x_n = c^1/n and the least value of the sum is n (c^1/n).

(iii) If x_l,x₂,…,x_n are variables and m_l,m₂,…,m_n are positive real number such that x_l + x₂ +….+ x_n = c (constant), then x_l^m_l * x₂^m₂ *… * x_n^m_n is greatest, when

x_l / m_l = x₂ / m₂ =…= x_n / m_n

= x_l + x₂ +….+ x_n / m_l + m₂ +….+ m_n