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 $\theta \in\left(0, \frac{\pi}{2}\right)$. Then the value of $\theta$ such that the sum of intercepts on axes made by this tangent is minimum is equal to :

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

Answer 1

Answer: (C)

Equation of tangent be
$\frac{ x \cos \theta}{3 \sqrt{3}}+\frac{ y \cdot \sin \theta}{1}=1, \quad \theta \in\left(0, \frac{\pi}{2}\right)$
intercept on $x$ -axis
$OA =3 \sqrt{3} \sec \theta$
intercept on $y$ -axis
$OB =\operatorname{cosec} \theta$
Now, sum of intercept
$=3 \sqrt{3} \sec \theta+\operatorname{cosec} \theta=f(\theta)$ let
$f^{\prime}(\theta)=3 \sqrt{3} \sec \theta \tan \theta-\operatorname{cosec} \theta \cot \theta$
$=3 \sqrt{3} \frac{\sin \theta}{\cos ^{2} \theta}-\frac{\cos \theta}{\sin ^{2} \theta}$
$=\underbrace{\frac{\cos \theta}{\sin ^{2} \theta} \cdot 3 \sqrt{3}}_{\oplus}\left[\tan ^{3} \theta-\frac{1}{3 \sqrt{3}}\right]=0 \Rightarrow \theta=\frac{\pi}{6}$

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