Question

Find the locus of the middle point of the chord of contact of tangents from any point of the circle $x^2+y^2=r^2$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Answer 1

Let any point on circle be $(r\cos \theta ,r\sin \theta )$
Then equation of chord of contact is
$\frac{x}{a^2}r\cos \theta -\frac{y}{b^2}r\sin \theta =1$ ....(i)
Let mid point of chord of contact is $(h,k)$
Then equation of chord of contact is
$\frac{hx}{a^2}-\frac{ky}{b^2}=\frac{h^2}{a^2}-\frac{k^2}{b^2}$ .....(ii)
On comparing (i) & (ii)
$\frac{r\cos \theta }{h}=\frac{r\sin \theta }{k}=\frac{1}{\frac{h^2}{a^2}-\frac{k^2}{b^2}}$
On solving we get required locus i.e.
${(\frac{x^2}{a^2}-\frac{y^2}{b^2})}^2=\frac{x^2+y^2}{r^2}$