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Question: Tangent is drawn to ellipse \(\frac{x^{2}}{27} + y^{2} = 1\)at \((3\sqrt{3}\cos\theta,\sin\theta)\l...

Tangent is drawn to ellipse x227+y2=1\frac{x^{2}}{27} + y^{2} = 1at

(33cos⁡θ,sin⁡θ)(whereθ∈(0,π2))(3\sqrt{3}\cos\theta,\sin\theta)\left( where\theta \in \left( 0,\frac{\pi}{2} \right) \right). Then the value of q such that sum of intercepts on axes made by this tangent is minimum, is

A

π3\frac{\pi}{3}

B

π6\frac{\pi}{6}

C

π8\frac{\pi}{8}

D

π4\frac{\pi}{4}

Answer

π6\frac{\pi}{6}

Explanation

Solution

The equation of tangent at given q point is

x.33cos⁡θ27+y.sin⁡θ1=1\frac{x.3\sqrt{3}\cos\theta}{27} + \frac{y.\sin\theta}{1} = 1

Ž Sum of the intercepts on axes is given by

Ž

dSdθ=33sec⁡θ⋅tan⁡θ−cosec⁡θ⋅cot⁡θ=0\frac { \mathrm { dS } } { \mathrm { d } \theta } = 3 \sqrt { 3 } \sec \theta \cdot \tan \theta - \operatorname { cosec } \theta \cdot \cot \theta = 0

Ž tan⁡3θ=133\tan^{3}\theta = \frac{1}{3\sqrt{3}} Ž tan⁡θ=13\tan\theta = \frac{1}{\sqrt{3}}Ž θ=π6\theta = \frac{\pi}{6}