Question

$P$ is the point on the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ and $Q$ is the corresponding point on the auxilliary circle of the ellipse. If the line joining centre $C$ to $Q$ meets the normal at $P$ with repsect to the given ellipse at $K,$ then the value of $CK$ is

Answer 1

Equation of normal at $P$ is $\frac{4x}{\cos \theta }-\frac{3y}{\sin \theta }=7\cdots \left(1\right)$

Equation of auxiliary circle is
$x^2+y^2=16$
Equation of normal at $Q$ will be
$y=\left(\tan \theta \right)x\cdots \left(2\right)$
from $\left(1\right)$ and $\left(2\right)$ point of intersection will be
$K\equiv (h,k)\equiv (7\cos \theta ,7\sin \theta )$
$\therefore CK=\sqrt{7^2({\sin }^2\theta +{\cos }^2\theta )}=7$ unit