Question

If the chord of contact of the tangents drawn from the points $(\alpha ,\beta )$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ touches the circles $x^2+y^2=c^2$, then the locus of the point $(\alpha ,\beta )$ is

• A. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{c^2}$
• B. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{1}{c^4}$
• C. $\frac{x^2}{a^4}+\frac{y^2}{b^4}=\frac{1}{c^2}$
• D. None of these

Answer 1

Answer: (C)

Let $(h,k)$ be the point then the equation of chord of contact for the ellipse
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is
$\frac{hx}{a^2}+\frac{k}{b^{2}y}-1=0$
Since it touch the circle $x^2+y^2=c^2$.
$\therefore \frac{(-1{)}^2}{\frac{h^2}{a^4}+\frac{k^2}{b^4}}=c^2$
$\Rightarrow \frac{x^2}{a^4}+\frac{y^2}{b^4}=\frac{1}{c^2}$