Discussion Forum

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

Answer:

Explanation:

  Equation of tangent at $(3\sqrt{3}\cos\theta, \sin \theta)$  on the ellipse  $\frac{x^{2}}{27}+\frac{y^{2}}{1}=1$ is

 $\frac{3\sqrt{3}x \cos \theta}{27}+\frac{y \sin \theta}{1}=1$

$\frac{x}{3\sqrt{3}} \cos \theta+\frac{y \sin \theta}{1}=1$

 Sum of intercepts of tangent

 i.e, $L=3\sqrt{3} \sec \theta+cosec \theta$

 $\because \frac{dL}{d \theta}=3\sqrt{3} \sec \theta \tan \theta- cosec \theta \cot \theta$

For  maxima or minima  $\frac{dL}{d \theta}$= 0

$3\sqrt{3} \sec \theta \tan \theta- cosec \theta \cot \theta=0$

$\tan ^{3} \theta = \frac{3}{3\sqrt{3}}\Rightarrow \tan \theta =1\sqrt{3}\Rightarrow \theta = \frac{\pi}{6}$

 Minimum   at $\theta$   = $\frac{\pi}{6}$