Mathematics - Ellipse Question with Solution | TestHub

The equation of the normal at $P(3\cos \theta ,2\sin \theta )$ is $3x\sec \theta -2y\mathrm{cosec}\theta =5$.

We are given that $\frac{5}{\sqrt{9{\sec }^2\theta +4{\mathrm{cosec}}^2\theta }}=\sqrt{3}$.

Squaring both sides, we get:

$$\frac{25}{9{\sec }^2\theta +4{\mathrm{cosec}}^2\theta }=3$$

$$25=3(9{\sec }^2\theta +4{\mathrm{cosec}}^2\theta )$$

$$9{\sec }^2\theta +4{\mathrm{cosec}}^2\theta =\frac{25}{3}$$

However, the minimum value of $9{\sec }^2\theta +4{\mathrm{cosec}}^2\theta$ is $25$.

Since $\frac{25}{3}<25$, no such $\theta$ exists.