A tangent is drawn at $(3\sqrt{3}\mathrm{cos\theta },\mathrm{sin\theta })\left(0<\mathrm{\theta }\frac{\mathrm{\pi }}{2}\right)$ to the ellipse $\frac{{\mathrm{x}}^2}{27}+\frac{{\mathrm{y}}^2}{1}=1.$ The value of $\mathrm{\theta }$ for which the sum of  the intercepts on the coordinate axes made by this tangent  attains the minimum is

Given ellipse $\frac{{\mathrm{x}}^2}{27}+\frac{{\mathrm{y}}^2}{1}=1$

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

 $$\begin{gathered} \frac{\mathrm{x}(3\sqrt{3} \mathrm{cos\theta })}{27}+\frac{\mathrm{y}(\sin \mathrm{\theta })}{1}=1 \\ \Rightarrow \frac{\mathrm{x}}{3\sqrt{3} \mathrm{sec\theta }}+\frac{\mathrm{y}}{\mathrm{cosec\theta }}=1 \end{gathered}$$

sum of intercepts $=3\sqrt{3} \mathrm{sec\theta }+\mathrm{cosec\theta }$

l$$\begin{gathered} \mathrm{Let} \mathrm{f}\left(\mathrm{\theta }\right) =3\sqrt{3} \mathrm{sec\theta }+\mathrm{cosec\theta } \\ Now {\mathrm{f}}^1\left(\mathrm{\theta }\right)=\frac{3\sqrt{3} {\sin }^3\mathrm{\theta }-{\cos }^3\mathrm{\theta }}{si{n}^2\mathrm{\theta } {\cos }^2\mathrm{\theta }} \end{gathered}$$

for max or min ${\mathrm{f}}^1\left(\mathrm{\theta }\right)=0$

$$\begin{gathered} \Rightarrow 3\sqrt{3} {\sin }^3\mathrm{\theta }={\cos }^3\mathrm{\theta } \\ \Rightarrow {\tan }^3\mathrm{\theta }=\frac{1}{3\sqrt{3}}={\left(\frac{1}{\sqrt{3}}\right)}^3 \\ \Rightarrow \mathrm{tan\theta }=\frac{1}{\sqrt{3}}=\tan \frac{\mathrm{\pi }}{6} \\ \Rightarrow \mathrm{\theta }=\frac{\mathrm{\pi }}{6} \end{gathered}$$