Exams Joint Entrance Examination Mathematics Trigonometric Ratios and Identities Question 98 Question 98 of 298 Medium MCQ
Let a tangent be drawn to the ellipse x 2 27 + y 2 = 1 \dfrac{x^2}{27}+y^2=1 27 x 2 + y 2 = 1 at ( 3 3 cos θ , sin θ ) (3\sqrt{3}\cos\theta,\sin\theta) ( 3 3 cos θ , sin θ ) , where θ ∈ ( 0 , π 2 ) \theta\in\left(0,\dfrac\pi2\right) θ ∈ ( 0 , 2 π ) . Then the value of θ \theta θ , such that the sum of intercepts on the axes made by this tangent is minimum, is equal to
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Tangent: x cos θ 3 3 + y sin θ = 1 \dfrac{x\cos\theta}{3\sqrt{3}}+y\sin\theta=1 3 3 x cos θ + y sin θ = 1 ; intercepts 3 3 sec θ 3\sqrt{3}\sec\theta 3 3 sec θ and csc θ \csc\theta csc θ .
\[\frac{d}{d\theta}\left(3\sqrt{3}\sec\theta+\csc\theta\right)=3\sqrt{3}\sec\theta\tan\theta-\csc\theta\cot\theta=0\ \Longrightarrow\ \tan^3\theta=\frac1{3\sqrt{3}}\ \Longrightarrow\ \tan\theta=\frac1{\sqrt{3}}.\]
\[\theta=\frac\pi6.\] Answer the question to unlock