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Question

The point of intersection of the tangents at the point P on the ellipsex2a2+y2b2=1 and its corresponding point Q on the auxiliary circle, lies on the line


A

x = a/e

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B

x = 0

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C

y = 0

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D

None of these

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Solution

The correct option is C

y = 0


Let the point P be (a cosθ, b sin θ) with eccentric angle θ. The corresponding point Q on the auxiliary circle will be (a cosθ, b sin θ).We know that

the equation of auxiliary circle of x2a2+y2b2=1 is x2 + y2 = a2 we will now find the tangents and find their intersection.

Tangent to ellipse:

(a cosθ, b sin θ) is the point ...(1)

tangent to circle:

(a cosθ, a sin θ) is the point

x2 + y2 = a2 or x2a2+y2b2=1 is the circle. ...(2)

(1)-(2)

y = 0

So all the tangents intersect on x - axis or y = 0.

Option (C).


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