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 :

Equation of tangent be

$\frac{x\cos \theta }{3\sqrt{3}}+\frac{y·\sin \theta }{1}=1, \theta \in \left(0,\frac{\pi }{2}\right)$

intercept on $x$-axis

$OA=3\sqrt{3}\sec \theta$

intercept on $y$-axis

$OB=cosec\theta$

Now, sum of intercept

$=3\sqrt{3}\sec \theta +cosec\theta =f\left(\theta \right)$ let

$f\text{'}\left(\theta \right)=3\sqrt{3}\sec \theta \tan \theta -cosec\theta \cot \theta$

$=3\sqrt{3}\frac{\sin \theta }{{\cos }^2\theta }-\frac{\cos \theta }{{\sin }^2\theta }$

$=\underset{\oplus }{\underbrace{\frac{\cos \theta }{{\sin }^2\theta }·3\sqrt{3}}}\left[{\tan }^3\theta -\frac{1}{3\sqrt{3}}\right]=0\Rightarrow \theta =\frac{\pi }{6}$

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