Let a tangent be drawn to the ellipse frac{x^ | vedclass

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(C) The equation of the tangent to the ellipse $\frac{x^{2}}{27}+y^{2}=1$ at the point $(3 \sqrt{3} \cos 	heta, \sin 	heta)$ is given by $\frac{x \c

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$\frac{\pi}{6}$

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(C) The equation of the tangent to the ellipse $\frac{x^{2}}{27}+y^{2}=1$ at the point $(3 \sqrt{3} \cos 	heta, \sin 	heta)$ is given by $\frac{x \cos 	heta}{3 \sqrt{3}}+\frac{y \sin 	heta}{1}=1$.The $x$-intercept is $OA = 3 \sqrt{3} \sec 	heta$ and the $y$-intercept is $OB = \operatorname{cosec} 	heta$.Let the sum of the intercepts be $f(	heta) = 3 \sqrt{3} \sec 	heta + \operatorname{cosec} 	heta$.To find the minimum,we differentiate $f(	heta)$ with respect to $	heta$:$f^{\prime}(	heta) = 3 \sqrt{3} \sec 	heta 	an 	heta - \operatorname{cosec} 	heta \cot 	heta$.Setting $f^{\prime}(	heta) = 0$:$3 \sqrt{3} \frac{\sin 	heta}{\cos^{2} 	heta} = \frac{\cos 	heta}{\sin^{2} 	heta}$$	an^{3} 	heta = \frac{1}{3 \sqrt{3}} = \left(\frac{1}{\sqrt{3}}ight)^{3}$$	an 	heta = \frac{1}{\sqrt{3}} \Rightarrow 	heta = \frac{\pi}{6}$.Since $f^{\prime}(	heta)  0$ for $	heta > \frac{\pi}{6}$,the function $f(	heta)$ attains its minimum value at $	heta = \frac{\pi}{6}$.

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 ..... .

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