Exams Joint Entrance Examination Mathematics Trigonometric Ratios and Identities Question 125 Question 125 of 298 Medium MCQ
If tangents are drawn to the ellipse x 2 + 2 y 2 = 2 x^2+2y^2=2 x 2 + 2 y 2 = 2 at all points on the ellipse other than its four vertices, then the mid-points of the tangents intercepted between the coordinate axes lie on the curve:
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Ellipse: x 2 2 + y 2 = 1 \dfrac{x^2}2+y^2=1 2 x 2 + y 2 = 1 ; tangent at ( 2 cos θ , sin θ ) (\sqrt{2}\cos\theta,\sin\theta) ( 2 cos θ , sin θ ) : x cos θ 2 + y sin θ = 1 \dfrac{x\cos\theta}{\sqrt{2}}+y\sin\theta=1 2 x cos θ + y sin θ = 1 .
Intercepts: P ( 2 cos θ , 0 ) P\left(\dfrac{\sqrt{2}}{\cos\theta},0\right) P ( cos θ 2 , 0 ) , Q ( 0 , 1 sin θ ) Q\left(0,\dfrac1{\sin\theta}\right) Q ( 0 , sin θ 1 ) ; midpoint ( h , k ) (h,k) ( h , k ) : h = 1 2 cos θ , k = 1 2 sin θ h=\dfrac1{\sqrt{2}\cos\theta},\ k=\dfrac1{2\sin\theta} h = 2 cos θ 1 , k = 2 sin θ 1 .
\[\frac1{2h^2}+\frac1{4k^2}=\cos^2\theta+\sin^2\theta=1.\] Answer the question to unlock