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Question

A straight line PQ touches the ellipse x216+y29=1 and the circle x2+y2=r2 (3<r<4). RS is chord of the circle which is parallel to PQ and passes through any focus of ellipse, then the length of RS is equal to unit.

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Solution

y=mx±16m2+9 is tangent to the ellipse x216+y29=1.

If this is also tangent to the circle x2+y2=r2, then
r2=16m2+91+m2m=r2916r2(3<r<4)(i)

Equation of line RS be
y0=r2916r2(x+7)

Length of chord drawn from centre C of circle to chord
=2r2d2
Where, d= perpendicular distance from centre to the chord
=∣ ∣ ∣ ∣ ∣ ∣7r2916r212+r2916r2∣ ∣ ∣ ∣ ∣ ∣
=r29

Hence, required length of chord =2r2(r29)2
=6 units.

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