Question

If the tangent and the normal to $x^2-y^2=4$ at a point cut off intercepts $a_1,a_2$ on the x-axis respectively and $b_1,b_2$ on the y-axis respectively, then the value of $a_1{a}_2+b_1{b}_2$ is

• A. $1$
• B. $-1$
• C. $0$
• D. $4$

Answer 1

The correct option is C $0$

Let any point on the given hyperbola is $P(a\sec \theta ,a\tan \theta )$
Therefore equation of tangent and normal to the hyperbola at point 'P' are given by,
$x\sec \theta -y\tan \theta =a$ and $\frac{x}{\sec \theta }+\frac{y}{\tan \theta }=2a$ respectively
$\therefore a_1=a\cos \theta ,b_1=-a\cot \theta$ and $a_2=2a\sec \theta ,b_1=2a\tan \theta$
Ergo $a_1{a}_2+b_1{b}_2=0$