Ques. -16

Geometry Level 4

Tangent is drawn to the ellipse x 2 27 + y 2 = 1 \frac{x^2}{27}+y^2=1 at ( 3 ⋅ 3 cos ⁡ θ , sin ⁡ θ ) (3 \cdot \sqrt{3} \cos\theta,\sin\theta) [ w h e r e θ ∈ ( 0 , π 2 ) ] [ where \theta \in \left( 0,\frac{\pi}{2}\right)] . then, the value of θ \theta such that the sum of intercepts on axes made by this tangent is minimum, is :

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π 6 \frac{\pi}{6} π 8 \frac{\pi}{8} π 4 \frac{\pi}{4} π 3 \frac{\pi}{3}

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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1 solution

Mudit Bansal
Feb 25, 2015

Equation of tangent at ( 3 3 cos ⁡ θ , sin ⁡ θ 3\sqrt { 3 } \cos { \theta } ,\sin { \theta } ) is: x 3 3 cos ⁡ θ + y sin ⁡ θ = 1 \frac { x }{ 3\sqrt { 3 } \cos { \theta } } +\frac { y }{ \sin { \theta } } =1 Sum of intercepts= 3 3 sec ⁡ θ + csc ⁡ θ 3\sqrt { 3 } \sec { \theta } +\csc { \theta } .Now,minimising this using differentiation we get tan ⁡ θ = 1 3 ⇒ θ = π 6 \tan { \theta } =\frac { 1 }{ \sqrt { 3 } } \Rightarrow \theta =\frac { \pi }{ 6 }

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