Question

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

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

Answer 1

The correct option is B $\frac{\pi }{6}$

Given ellipse is $\frac{x^2}{27}+y^2=1$
Now tangent at $(3\sqrt{3}\cos \theta ,\sin \theta )$ is
$T=0\Rightarrow \frac{x\cos \theta }{3\sqrt{3}}+y\sin \theta =1\Rightarrow \frac{x}{\frac{3\sqrt{3}}{\cos \theta }}+\frac{y}{\frac{1}{\sin \theta }}=1$
Sum of intercepts
$S=\frac{3\sqrt{3}}{\cos \theta }+\frac{1}{\sin \theta }\Rightarrow S=3\sqrt{3}\sec \theta +\text{cosec}\theta$
Now,
$y^{\prime }=3\sqrt{3}\sec \theta \tan \theta -\text{cosec}\theta \cot \theta y^{\prime \prime }=3\sqrt{3}{\sec }^2\theta \tan \theta +3\sqrt{3}{\sec }^3\theta +{\text{cosec}}^2\theta \cot \theta +{\text{cosec}}^3\theta$
For minima/maxima
$y^{\prime }=0\Rightarrow 3\sqrt{3}{\tan }^3\theta =1\Rightarrow \tan \theta =\frac{1}{\sqrt{3}}\Rightarrow \theta =\frac{\pi }{6}$
Here $y^{\prime \prime }>0$
So, minimum occur at $\theta =\frac{\pi }{6}$