Question

A tangent is drawn to the ellipse $x^2+27y^2=27$ at a point $P\left(\theta \right)$ where $\theta \in (0,\frac{\pi }{2}).$ Then the minimum sum of the intercepts made by the tangent at $P$ on the co-ordinate axes is equal to

Answer 1

Equation of ellipse is $\frac{x^2}{27}+\frac{y^2}{1}=1$
Equation of tangent at $P\left(\theta \right)$ is
$\frac{x}{3\sqrt{3}}\cos \theta +\frac{y}{1}\sin \theta =1$
$\Rightarrow \frac{x}{3\sqrt{3}\sec \theta }+\frac{y}{\text{cosec}\theta }=1$
Now, sum of intercepts,
$f\left(\theta \right)=3\sqrt{3}\sec \theta +\text{cosec}\theta$
$\Rightarrow f^{\prime }\left(\theta \right)=3\sqrt{3}\sec \theta \tan \theta -\text{cosec}\theta \cot \theta$
$=\frac{3\sqrt{3}{\sin }^3\theta -{\cos }^3\theta }{{\cos }^2\theta \cdot {\sin }^2\theta }$
$f^{\prime }\left(\theta \right)=0$
$\Rightarrow \tan \theta =\frac{1}{\sqrt{3}}$
$\Rightarrow \theta =\frac{\pi }{6}$
$\Rightarrow f^{\prime }\left({\frac{\pi }{6}}^{-}\right)<0$ and $f^{\prime }\left({\frac{\pi }{6}}^{+}\right)>0$
$\Rightarrow f\left(\theta \right)$ will have local minima at $\theta =\frac{\pi }{6}$
$f_{min}\left(\theta \right)=3\sqrt{3}\left(\frac{2}{\sqrt{3}}\right)+2=8$