Question

The locus pf point of intersection of tangents at the end of normal chord of hyperbola $x^2-y^2=a^2$ is :

• A. $a^2(y^2-x^2)=4x^2{y}^2$
• B. $a^2(y^2+x^2)=4x^2{y}^2$
• C. $y^2+x^2=4x^2{y}^2$
• D. None of these

Answer 1

The correct option is A $a^2(y^2-x^2)=4x^2{y}^2$

Let the point be $(x_1,y_1),$

Equation of chord of contact of tangents drawn from the point $(x_1,y_1)$ to the hyperbola $x^2-y^2=a^2$ is ${xx}_1-{yy}_1=a^2$ ...(1)

Equation pf normal chord is $\frac{x}{\sec \theta }+\frac{y}{\tan \theta }=2a$ ...(2)

Since (1) and (2) are identical, comparing coefficients in (1) and (2), we get $\frac{x_1}{\frac{1}{\sec \theta }}=\frac{-y_1}{\frac{1}{\tan \theta }}=\frac{a^2}{2a}$

$\Rightarrow \sec \theta -\frac{a}{2x_1}$ and $\tan \theta =\frac{-a}{2y_1}$

$\Rightarrow {\sec }^2\theta -{\tan }^2\theta =1$

$\Rightarrow \frac{a^2}{4x_1^2}-\frac{a^2}{4y_1^2}=1$

$\therefore$ The required locus is $a^2(y^2-x^2)=4x^2{y}^2.$