If the straight line x cos α + y sin α = p touches the curve x 2 a 2 - y 2 b 2 = 1 , then prove that a 2 cos 2 α - b 2 sin 2 α = p 2 .

We have, x cos α + y sin α = p . . . i x 2 a 2 - y 2 b 2 = 1 . . . ii As, the straight line i touches the curve ii . So, the straight line i is tangent to the curve ii . Also, the slope of the straight line, m = - cos α sin α And, the slope of the tangent to the curve = dy dx = b 2 a 2 × x y So, b 2 a 2 × x y = - cos α sin α ⇒ xb 2 sin α = - ya 2 cos α ⇒ x = - ya 2 cos α b 2 sin α . . . iii Substituting the value of x in i , we get x cos α + y sin α = p ⇒ - ya 2 cos 2 α b 2 sin α + y sin α = p ⇒ - ya 2 cos 2 α + yb 2 sin α b 2 sin α = p ⇒ y - a 2 cos 2 α + b 2 sinα = pb 2 sin α ⇒ y = pb 2 sin α b 2 sin α - a 2 cos 2 α So, from iii , we get x = - ya 2 cos α b 2 sin α = - ya 2 cos α b 2 sin α = - a 2 cos α b 2 sin α × pb 2 sin α b 2 sin α - a 2 cos 2 α ⇒ x = - pa 2 cos α b 2 sin α - a 2 cos 2 α Substituting the values x and y in ii , we get x 2 a 2 - y 2 b 2 = 1 ⇒ 1 a 2 × - pa 2 cos α b 2 sin α - a 2 cos 2 α 2 - 1 b 2 × pb 2 sin α b 2 sin α - a 2 cos 2 α 2 = 1 ⇒ p 2 a 2 cos 2 α b 2 sin α - a 2 cos 2 α 2 - p 2 b 2 sin 2 α b 2 sin α - a 2 cos 2 α 2 = 1 ⇒ p 2 a 2 cos 2 α - p 2 b 2 sin 2 α b 2 sin α - a 2 cos 2 α 2 = 1 ⇒ p 2 a 2 cos 2 α - b 2 sin 2 α b 2 sin α - a 2 cos 2 α 2 = 1 ⇒ p 2 a 2 cos 2 α - b 2 sin 2 α a 2 cos 2 α - b 2 sin 2 α 2 = 1 ⇒ p 2 a 2 cos 2 α - b 2 sin 2 α = 1 ∴ p 2 = a 2 cos 2 α - b 2 sin 2 α .

E M B I B E

MATHEMATICS CLASS XII VOLUME-1>Chapter 16 - Tangents and Normals>EXERCISE

R D Sharma Solutions for Chapter: Tangents and Normals, Exercise 1: EXERCISE

Author:R D Sharma

R D Sharma Mathematics Solutions for Exercise - R D Sharma Solutions for Chapter: Tangents and Normals, Exercise 1: EXERCISE

Attempt the free practice questions on Chapter 16: Tangents and Normals, Exercise 1: EXERCISE with hints and solutions to strengthen your understanding. MATHEMATICS CLASS XII VOLUME-1 solutions are prepared by Experienced Embibe Experts.

Questions from R D Sharma Solutions for Chapter: Tangents and Normals, Exercise 1: EXERCISE with Hints & Solutions

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 16 - Tangents and Normals>EXERCISE>Q 2

Show that the following set of curve intersect orthogonally $y = x^{3}$ and $18 y = 21 - 2 x^{3}$

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 16 - Tangents and Normals>EXERCISE>Q 2

Show that the following set of curve intersect orthogonally $x^{3} - 3 {xy}^{2} = - 2$ and $3 x^{2} y - y^{3} = 2$ ?

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 16 - Tangents and Normals>EXERCISE>Q 4

Show that the curves $4 x = y^{2}$ and $4 xy = k$ cut at right angles, if $k^{2} = 512$ ?

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 16 - Tangents and Normals>EXERCISE>Q 5

Show that the curves $2 x = y^{2}$ and $2 xy = k$ cut at right angles, if $k^{2} = 8$ ?

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 16 - Tangents and Normals>EXERCISE>Q 6

Prove that the curves $xy = 4$ and $x^{2} + y^{2} = 8$ touch each other.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 16 - Tangents and Normals>EXERCISE>Q 7

Prove that the curves $y^{2} = 4 x$ and $x^{2} + y^{2} - 6 x + 1 = 0$ touch each other at the point $(1, 2)$ .

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 16 - Tangents and Normals>EXERCISE>Q 8

Find the condition for the following set of curve to intersect orthogonally $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and $\frac{x^{2}}{A^{2}} - \frac{y^{2}}{B^{2}} = 1$ .

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 16 - Tangents and Normals>EXERCISE>Q 10

If the straight line $x \cos α + y \sin α = p$ touches the curve $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ , then prove that $a^{2} \cos^{2} α - b^{2} \sin^{2} α = p^{2}$ .

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