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

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