Question

The chords of contact of a point $P$ w.r.t. a hyperbola and its auxiliary circle are at right angle, then the point $P$ lies on :

• A. conjugate hyperbola
• B. one of the directirx
• C. one of the asymptotes
• D. None of these

Answer 1

Answer: (C)

one of the asymptotes

Let $P$ be $(h,k)$ be any point. The chord of contact of $P$ w.r.t. the hyperbola is $\frac{hx}{a^2}-\frac{ky}{b^2}=1$ ...(i)

Slope of chord of contact to hyperbola $=\frac{h{b}^2}{k{a}^2}$
The chord of contact of $P$ w.r.t. the auxiliary circle is
$hx+ky=a^2-b^2$ ....(ii)
Slope of chord of contact w.r.t. the auxiliary circle $=-\frac{h}{k}$
Now, $\frac{h{b}^2}{k{a}^2}\times (-\frac{h}{k})=-1$
$\Rightarrow h^2{b}^2-k^2{a}^2=0$
$\Rightarrow \frac{h^2}{a^2}-\frac{k^2}{b^2}=0$
Therefore, $P$ lies on one of the asymptotes.