Question

If the chords of contact of tangents drawn from $P$ to the hyperbola $x^2-y^2=a^2$ and its auxiliary circle are at right angle, then $P$ lies on :

• A. $x^2-y^2=3a^2$
• B. $x^2-y^2=2a^2$
• C. $x^2-y^2=0$
• D. $x^2-y^2=1$

Answer 1

The correct option is C $x^2-y^2=0$

Let $P$ be $(h,k)$
Now Chord of contact of tangent from $P$ to the hyperbola $x^2-y^2=a^2$ is,
$T=0\Rightarrow hx-ky=a^2$ (i)
And director circle of given hyperbola is, $x^2+y^2=a^2$
Thus equation of chord of contact to this circle from P is, $hx+ky=a^2$ (ii)
Now given line (i) and (ii) are perpendicular,
$\Rightarrow \frac{h}{k}\times \frac{-h}{k}=-1\Rightarrow h^2=k^2$
Hence locus of $P$ is given by, $x^2-y^2=0$