Tangent is drawn to ellipse   $\frac{x^2}{27}+y^2=1at(3\sqrt{3}\cos \theta ,\sin \theta )$ $(\text{where\hspace{0.17em}}\theta \in (0,\pi /2\left)\right)$ . Then the value of such that sum of intercepts on 
 axes made by this tangent is least is 

$\text{ Equation of tangent at }(3\sqrt{3}\cos \theta ,\sin \theta )\text{ is }$$\frac{x\cos \theta }{3\sqrt{3}}+y\sin \theta =1$

Its intercept on coordinates axes are  $3\sqrt{3}/\cos \theta and\cos ec\theta$ If S denotes their sum, then

$\begin{array}{l}S=3\sqrt{3}\sec \theta +\mathrm{cosec}\theta \\ \frac{dS}{d\theta }=3\sqrt{3}\sec \theta \tan \theta -\mathrm{cosec}\theta \cot \theta \\ \text{ For minimum value of }S\cdot \frac{dS}{d\theta }=0\\ \Rightarrow \frac{3\sqrt{3}\sin \theta }{{\cos }^2\theta }-\frac{\cos \theta }{{\sin }^2\theta }=0\Rightarrow {\tan }^3\theta =3\sqrt{3}\end{array}$

$\begin{array}{l}\text{ Since }\theta \in (0,\pi /2),\text{ so }\tan \theta =\sqrt{3}\Rightarrow \theta =\pi /3\\ \text{ Also }\frac{d^{2}S}{d{\theta }^2}=3\sqrt{3}\sec \theta {\tan }^2\theta +3\sqrt{3}{\sec }^3\theta +{\mathrm{cosec}}^2\theta \\ \cot \theta +{\mathrm{cosec}}^3\theta \\ {\left.\Rightarrow \frac{d^{2}S}{d{\theta }^2}\right|}_{\theta -\pi /3}>0.\underset{\_}{\text{ Thus }\mathrm{S}}\text{ is least when }\theta =\pi /3\end{array}$