Question
Tangent is drawn to ellipse
$$\frac{{{x^2}}}{{27}} + {y^2} = 1$$ at $$\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)$$ (where $$\theta \in \left( {0,\frac{\pi }{2}} \right)$$ ).
Then the value of $$\theta $$ 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 :
$$\frac{\pi }{8}$$
Solution :
$$\eqalign{
& {\text{Equation}}\,{\text{of}}\,{\text{tangent}}\,{\text{to}}\,{\text{the}}\,{\text{ellipse}}\,\frac{{{x^2}}}{{27}} + {y^2} = 1\,{\text{at}} \cr
& \left( {3\sqrt 3 \cos \theta ,\sin \theta } \right),\theta \in \left( {0,\frac{\pi }{2}} \right)\,{\text{is}}\,\frac{{\sqrt 3 x\cos \theta }}{9} + y.\sin \theta = 1 \cr
& \therefore {\text{Intercept}}\,{\text{on}}\,x - {\text{axis}}\, = \frac{9}{{\sqrt 3 \cos \theta }}; \cr
& {\text{Intercept}}\,{\text{on}}\,y - {\text{axis}} = \frac{1}{{\sin \theta }} \cr
& \therefore {\text{Sum}}\,{\text{of}}\,{\text{intercepts}} = S = 3\sqrt 3 \sec \theta + \operatorname{cosec} \theta \cr
& {\text{For}}\,{\text{min}}{\text{.}}\,{\text{value}}\,{\text{of}}\,S,\frac{{dS}}{{d\theta }} = 0 \cr
& \Rightarrow 3\sqrt 3 \sec \theta \tan \theta - \operatorname{cosec} \theta \cot \theta = 0 \cr
& \Rightarrow \frac{{3\sqrt 3 \sin \theta }}{{{{\cos }^2}\theta }} - \frac{{\cos \theta }}{{{{\sin }^2}\theta }} = 0 \Rightarrow 3\sqrt 3 {\sin ^3}\theta - {\cos ^3}\theta = 0 \cr
& \Rightarrow {\tan ^3}\theta = \frac{1}{{3\sqrt 3 }} = {\left( {\frac{1}{{\sqrt 3 }}} \right)^3} \cr
& \Rightarrow \tan \theta = \frac{1}{{\sqrt 3 }} = \tan \frac{\pi }{6} \Rightarrow \theta = \frac{\pi }{6} \cr} $$