Let a tangent be drawn to the ellipse {{{x^2}} \over {27}} + {y^2} = 1 at (3\sqrt 3 \cos heta ,\sin heta ) where 0 \in \left( {0,{\pi \over 2}} ight). Then the value of heta such that the sum of intercepts on axes made by this tangent is minimum is equal to :

Question

Let a tangent be drawn to the ellipse {{{x^2}} \over {27}} + {y^2} = 1 at (3\sqrt 3 \cos 	heta ,\sin 	heta ) where 0 \in \left( {0,{\pi \over 2}} ight). Then the value of 	heta such that the sum of intercepts on axes made by this tangent is minimum is equal to :

Accepted Answer

Answer: A. {{\pi \over 6}} Tangent = {x \over {3\sqrt 3 }}\cos 	heta + y\sin 	heta = 1 x-intercept = {3\sqrt 3 } sec	heta y-intercept = cosec	heta sum = {3\sqrt 3 } sec	heta + cosec	heta = f(	heta) 	heta\in\left( {0,{\pi \over 2}} ight) \Rightarrow f'(	heta) = {3\sqrt 3 } sec	hetatan	heta - cosec	heta cot	heta = 0 \Rightarrow {{3\sqrt 3 \sin 	heta } \over {{{\cos }^2}	heta }} = {{\cos 	heta } \over {\sin 	heta }} \Rightarrow {	an ^3}	heta = {\left( {{1 \over {\sqrt 3 }}} ight)^3} \Rightarrow 	an 	heta = {1 \over {\sqrt 3 }} \Rightarrow 	heta = {{\pi \over 6}} also f'(	heta) changes sign - to + hence minimum.

Let a tangent be drawn to the ellipse $${{{x^2}} \over {27}} + {y^2} = 1$$ at $$(3\sqrt 3 \cos \theta ,\sin \theta )$$ where $$0 \in \left( {0,{\pi \over 2}} \right)$$. Then the value of $$\theta$$ such that the sum of intercepts on axes made by this tangent is minimum is equal to :

Answer

Explanation

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