OP Malhotra Class-11 Hyperbola S.Chand ISC Maths Solution

OP Malhotra Class-11 Hyperbola S.Chand ISC Maths Solution Chapter-25. Step by step Solutions of OP Malhotra S.Chand ISC Class-11 Mathematics with Exe-25 (a), 25 (b), With Chapter Test. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

OP Malhotra Class-11 Hyperbola S.Chand ISC Maths Solution

Class:	11th
Subject:	Mathematics
Chapter  :	Ch-25 Hyperbola of Section -B

Board	ISC
Writer	OP Malhotra
Publications	S.Chand Publications 2020-21

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–: Select Topics :-

Exe-25 (a)

Exe-25 (b)

Chapter Test

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OP Malhotra Class-11 Hyperbola S.Chand ISC Maths Solution

Hyperbola
A hyperbola is the locus of a point in a plane which moves in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed line is always constant which is always greater than unity. The fixed point is called the focus, the fixed line is called the directrix and the constant ratio, generally denoted bye, is known as the eccentricity of the hyperbola.

Diameter and Conjugate Diameter

1. Diameter The locus of the mid-points of a system of parallel chords of a hyperbola is called a diameter.>
   The equation of the diameter bisecting a system of parallel chord of slope m to the hyperbola  is
2. Conjugate Diameter The diameters of a hyperbola are sal to be conjugate diameter, if each bisect the chords parallel to th other.
   The diameters y = m₁x and y = m₂x are conjugate, if m₁ m₂ = b²/a².
3. In a pair of conjugate diameters of a hyperbola, only one mee the hyperbola in real points.

Asymptote

An asymptote to a curve is a straight line, at a finite distance from the origin, to which the tangent to a curve tends as the point of contact goes to infinity.

1. The equation of two asymptotes of the hyperbola  are
2. The combined equation of the asymptotes to the hyperbola
3. When b = a, i.e., the asymptotes of rectangular hyperbola x² – y² = a² are y = ± x which are at right angle.
4. A hyperbola and its conjugate hyperbola have the same asymptotes.
5. The equation of the pair of asymptotes differ the hyperbola and the conjugate hyperbola by the same constant only i.e., Hyperbola — Asymptotes = Asymptotes — Conjugate hyperbola
6. The asymptotes pass through the centre of the hyperbola.
7. The bisectors of angle between the asymptotes are the coordinate axes.
8. The angle between the asymptotes of  is 2 tan^-1(b/a) or 2 sec^-1(e).

Rectangular Hyperbola

A hyperbola whose asymptotes include a right angle is said to I rectangular hyperbola or we can say that, if the lengths of transver: and conjugate axes of any hyperbola be equal, then it is said to be rectangular hyperbola.

Tangent Equation of Rectangular Hyperbola xy = c²

1. Point Form The equation of tangent at (x₁, y₁) to the rectangular hyperbola is xy₁ + yx₁ = 2c² or (x/x₁ + y/y₁) = 2.
2. Parametric Form The equation of tangent at (ct, c/t) to the hyperbola is( x/t + yt) = 2c.
3. Tangent at P(ct₁, c/t₁) and Q (ct₂, c/t₂) to the rectangular hyperbola intersect a
4. The equation of the chord of contact of tangents drawn from a point (x₁, y₁) to the rectangular hyperbola is xy₁ + yx₁ = 2c².

Normal Equation of Rectangular Hyperbola xy = c²

1. Point Form The equation of the normal at (x₁, y₁) to the rectangular hyperbola is xx₁ – yy₁ = x₁² – y₁².
2. Parametric Form The equation of the normal at ( ct, c/t)to the rectangular hyperbola xy = c² is xt³ — yt — ct⁴ + c = O.
3. The equation of the normal at( ct, c/t)is a fourth degree equation t in t. So, in general four normals can be drawn from a point to the hyperbola xy = c².

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Exe-25 (a)

OP Malhotra Class-11 Hyperbola S.Chand ISC Maths Solution

Page 25-11 to 25-12

Question 1:

Find the equation of the hyperbola whose focus is (1, 1) and direction ………………… √2.

Question 2:

………………………

………………………

………………………

Question 12:

Find the focus of the point such that the difference ………………………… Name the curve.

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Exe-25 (b)

OP Malhotra Class-11 Hyperbola S.Chand ISC Maths Solution

Page 25-18

Question 1:

Find the tangent to the parabola  ……………… with the x-axis.

Question 2:

………………………

………………………

…………………………

Question 16:

Find the equation of the tangent …………………. to the line 6x + 5y = 21.

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Chapter Test

OP Malhotra Class-11 Hyperbola S.Chand ISC Maths Solution

Page 25-22

-: End of Hyperbola Solution :-

Return to :-  OP Malhotra S. Chand ISC Class-11 Maths Solutions

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