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Question

From a point P on the circle C1:x2+y2=25, tangents are drawn to the circle C2:x2+y2=9. Let θ be the angle made by the line PO (O is the origin) with positive x-axis. If the chord of contact of the circle C2 is tangent to another circle C3 which passes through the origin, then locus of center of the circle C3 is

A
a parabola with focus at (0,3sinθ) and for θ=0, vertex is (9320,0)
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B
a parabola with focus at (0,0) and for θ=π4, vertex is (9220,9220)
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C
a parabola with focus at (0,0) and for θ=π4, vertex is (910,0)
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D
a parabola with focus at (0,0) and for θ=0, vertex is (0,910)
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Solution

The correct option is B a parabola with focus at (0,0) and for θ=π4, vertex is (9220,9220)
Any point on the circle C1 is P(5cosθ,5sinθ)
Chord of contact of the tangents drawn from the P is
5xcosθ+5ysinθ=9xcosθ+ysinθ=95

Let the center of C3 is (h,k)
If the radius of C3 is r,
then h2+k2=r
and |5hcosθ+5ksinθ9|5=r
locus of the point (h,k) is a parabola whose focus is at the origin and xcosθ+ysinθ=95 as directrix.

For θ=π4, h=k|5h+5k92|52=h2+k25h+5k92=52(h2+k2)(centre and origin lie on the same side in the 1st quadrant)
(h,k)(9220,9220)

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