Question

P is a point on the circle $x^2+y^2=c^2$. The locus of the mid-points of chords of contact of P with respect to $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, is:

• A. $c^2(\frac{x^2}{a^2}+\frac{y^2}{b^2})=x^2+y^2$
• B. $c^2{(\frac{x^2}{a^2}+\frac{y^2}{b^2})}^2=x^2+y^2$
• C. $c^2(\frac{x^2}{a^2}+\frac{y^2}{b^2})=(x^2+y^2{)}^2$
• D. None of these

Answer 1

The correct option is A $c^2(\frac{x^2}{a^2}+\frac{y^2}{b^2})=x^2+y^2$

Let $P(c\cos \theta ,c\sin \theta )$
equation of polar is $\frac{cx\cos \theta }{a^2}+\frac{cy\sin \theta }{b^2}=1$ ...(1)
let $(h,k)$ be the mid point
Then equation of chord is $T=S_1$
$\Rightarrow \frac{hx}{a^2}+\frac{ky}{b^2}=\frac{h^2}{a^2}+\frac{k^2}{b^2}$ ..(2)
Comparing (1) and (2): $\frac{h}{c\cos \theta }=\frac{k}{c\sin \theta }=\frac{h^2}{a^2}+\frac{k^2}{b^2}$
Eliminating $\theta$ : $c^2{(\frac{h^2}{a^2}+\frac{k^2}{b^2})}^2=h^2+k^2$