Question

From the points on the circle $x^2+y^2=a^2$, tangents are drawn to the hyperbola $x^2-y^2=a^2$; the locus of the middle points of the chords of contact is

• A. $\frac{x^2-y^2}{a^2{x}^2+y^2}=a^2$
• B. ${(x^2-y^2)}^2=a^2(x^2+y^2)$
• C. ${(x^2+y^2)}^2=a^2(x^2-y^2)$
• D. $x^2+y^2=a^2(x^2-y^2)$

Answer 1

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

Let point on the circle is $P(a\cos \theta ,a\sin \theta )$
So equation of chord of contact of tangent from point $P$ is, $T=0$
$x\cos \theta -y\sin \theta =a..\left(1\right)$
Let mid point of chord is $R(h,k)$
Thus equation of chord with mid point as $R$ is, $T=S_1$
$\Rightarrow hx-ky=h^2-k^2..\left(2\right)$
But both line are same,
$\Rightarrow \frac{h}{\sin \theta }=\frac{k}{\sin \theta }=\frac{h^2-k^2}{a}$
$\Rightarrow \cos \theta =\frac{ah}{h^2-k^2},\sin \theta =\frac{ak}{h^2-k^2}$
Eliminating $\theta$ we get,
$(h^2-k^2{)}^2=a^2(h^2+k^2)$
Therefore, required locus of Point $R$ is, $(x^2-y^2{)}^2=a^2(x^2+y^2)$
Hence. option 'B' is correct.