Question

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

• A. $\pi /3$
• B. $\pi /6$
• C. $\pi /8$
• D. $\pi /4$

Answer 1

The correct option is B $\pi /6$

Equation of tangent at $(3\sqrt{3}\cos \theta ,\sin \theta )$

$\frac{n\left(3\sqrt{3}\cos \theta \right)}{27}+\frac{y\left(\sin \theta \right)}{1}=1$

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

$\therefore$ sum of intersects on axes $=3\sqrt{3}\sec \theta +\csc \theta =f\left(\theta \right)$

$\therefore f^{‘}\left(\theta \right)=\frac{3\sqrt{3}{\sin }^3\theta -{\cos }^3\theta }{{\sin }^2\theta .{\cos }^2\theta }$

For $f\left(\theta \right)$ to be minimum $f^{‘}\left(\theta \right)=0$

$\Rightarrow 3\sqrt{3}{\sin }^3\theta ={\cos }^3\theta$

$\Rightarrow \tan \theta =\frac{1}{\sqrt{3}}$

$\theta =\frac{\pi }{6}$

So option $B$ is correct.