Question

Tangent is drawn to ellipse $\frac{x^2}{27}+y^2=1$ at ($3\sqrt{3}\cos \theta ,\sin \theta$) (where $$\theta \in (0,\pi/2)$.Thenthevalueof$ \theta $ such that sum of intercepts on axes made by this tangent is minimum, is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{6}$
• C. $\frac{\pi }{8}$
• D. $\frac{\pi }{4}$

Answer 1

The correct option is B $\frac{\pi }{6}$

$\frac{x}{3\sqrt{3}}\cos \theta +\frac{4}{3}\sin \theta =1$

$x=0,y=\csc \theta$

$y=0,x=3\sqrt{3}\sec \theta$

$3\sqrt{3}\sec \theta +\csc \theta \to min$

$f\left(\theta \right)=3\sqrt{3}\sec \theta +\csc \theta$

$f\left(\theta \right)=0=3\sqrt{3}\sec \theta \tan \theta -\csc \theta \cot \theta$

$\frac{3\sqrt{3}\sin \theta }{{\cos }^2}=0\Rightarrow 3\sqrt{3}\sin 3\theta -\cos 3\theta =0$

$(\sqrt{3}\sin \theta {)}^3-(\cos \theta {)}^3=0$

$(\sqrt{3}{)}^3=(\cot \theta {)}^3$

$\cot \theta =\sqrt{3}\Rightarrow \theta =\pi /6$