Tangent is drawn to ellipse frac{{{x^2}}}{{27}} + {y^2} = 1,at,(3√ 3 cos θ ,sin θ ) where

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10-2. Parabola, Ellipse, Hyperbola

normal

Tangent is drawn to ellipse $\frac{{{x^2}}}{{27}} + {y^2} = 1\,at\,(3\sqrt 3 \cos \theta ,\sin \theta )$ where $\theta \in (0, \pi /2)$ . Then the value of $\theta$ such that sum of intercepts on axes made by this tangent is minimum, is

$\pi /3$

$\pi /6$

$\pi /8$

$\pi /4$

$\frac{x \cos \theta}{3 \sqrt{3}}+y \sin \theta=1.$

Sum of intercepts $ = 3\sqrt 3 \sec \theta + \cos ec\theta = {\rm{f}}(\theta )$ say

$f'\,(\theta ) = \frac{{3\sqrt 3 {{\sin }^3}\theta – {{\cos }^3}\theta }}{{{{\sin }^2}\theta {{\cos }^2}\theta }}$

At $\theta=\frac{\pi}{6}, f(\theta)$ is minimum

Standard 11

Mathematics

medium

hard

(IIT-2003)

normal

medium

(JEE MAIN-2020)

hard

(IIT-1999)