Question

Any ordinate $MP$ of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ meets the auxiliary circle at $Q$, then locus of the point of intersection of normals at $P$ and $Q$ to the respective curves is

• A. $x^2+y^2=8$
• B. $x^2+y^2=34$
• C. $x^2+y^2=64$
• D. $x^2+y^2=16$

Answer 1

The correct option is C $x^2+y^2=64$

Equation of given ellipse is
$\frac{x^2}{25}+\frac{y^2}{9}=1$
Let $P\equiv (5\cos \theta ,3\sin \theta ),$ then coordinates of $Q$ are $(5\cos \theta ,5\sin \theta ).$
Equation of normal to the ellipse at $P$ is
$5x\sec \theta -3y\text{cosec}\theta =16\dots \left(i\right)$
Equation of normal to the circle $x^2+y^2=25$ at point $Q$ is
$y=x\tan \theta \dots \left(ii\right)$
$\therefore$Finding intersection point and eliminating $\theta$ from $\left(i\right)$ and $\left(ii\right)$, we get $x^2+y^2=64.$