Question

Any ordinate $MP$ of ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ meets the auxiliary circle at $Q$, then locus of point of intersection of normals at $P$ and $Q$ to curves is

• A. $x^2+y^2=8$
• B. $x^2+y^2=34$
• C. $x^2+y^2=64$
• D. $x^2+y^2=81$

Answer 1

The correct option is D $x^2+y^2=81$

Normal to ellipse at $P$ is
$\frac{5x}{\cos \theta }-\frac{4y}{\sin \theta }=\mathrm{9...}\left(i\right)$
equation of normal to circle at $Q$ is
$y=x\tan \theta ....\left(ii\right)$
eliminating $\theta$ from (i) and (ii) we get
$x^2+y^2=81$