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

Tangent = ${x \over {3\sqrt 3 }}\cos \theta + y\sin \theta = 1$<br><br>x-intercept = ${3\sqrt 3 }$ sec$\theta$<br><br>y-intercept = cosec$\theta$<br><br>sum = ${3\sqrt 3 }$ sec$\theta$ + cosec$\theta$ = f($\theta$) $\theta$$\in$$\left( {0,{\pi \over 2}} \right)$<br><br>$\Rightarrow$ f'($\theta$) = ${3\sqrt 3 }$ sec$\theta$tan$\theta$ $-$ cosec$\theta$ cot$\theta$ = 0<br><br>$\Rightarrow$ $${{3\sqrt 3 \sin \theta } \over {{{\cos }^2}\theta }} = {{\cos \theta } \over {\sin \theta }}$$<br><br>$\Rightarrow {\tan ^3}\theta = {\left( {{1 \over {\sqrt 3 }}} \right)^3}$<br><br>$\Rightarrow \tan \theta = {1 \over {\sqrt 3 }}$<br><br>$\Rightarrow$ $\theta$ = ${{\pi \over 6}}$<br><br>also f'($\theta$) changes sign $-$ to + hence minimum.