Question

The tangent at $(3\sqrt{3}\cos \theta ,\sin \theta )$ is drawn to the ellipse $\frac{x^2}{27}+y^2=1$. Then the value of $\theta$ such that the sum of intercepts on axes made by the tangent is minimum is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{6}$
• C. $\frac{\pi }{8}$
• D. $\frac{\pi }{4}$

Answer 1

Answer: (B)

$\frac{\pi }{6}$

$\frac{x\cos \theta }{3\sqrt{3}}+y\sin \theta =1$
Sum of intercepts $=3\sqrt{3}\sec \theta +\text{cosec}\theta =f\left(\theta \right)$ (let)
$f^{\prime }\left(\theta \right)=\frac{3\sqrt{3}{\sin }^3\theta -{\cos }^3\theta }{{\sin }^2\theta {\cos }^2\theta }$
For $\theta =\frac{\pi }{6},f\left(\theta \right)$ is minimum