Let a tangent be drawn to the ellipse x ^ 2 27 + y ^ 2 = 1 at ( 3 3...

Equation of tangent be

$\dfrac{xcos\theta}{3\sqrt{3}} + \dfrac{y \cdot sin\theta}{1} = 1, \theta \in \left(0, \dfrac{\pi}{2}\right)$

intercept on $x$-axis

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

intercept on $y$-axis

$OB = cosec \theta$

Now, sum of intercept

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

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

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

$= \mathop{\mathop{\dfrac{cos\theta}{\sin^{2}\theta} \cdot 3\sqrt{3}}_{⏟}}_{⊕}\left[\tan^{3}\theta - \dfrac{1}{3\sqrt{3}}\right] = 0 ⇒ \theta = \dfrac{\pi}{6}$

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