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Question

Tangents are drawn one to each of the concentric circles x2+y2=4 and x2+y2=9. They include an angle of 60. Find the locus of the point of intersection of these tangents.

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Solution

Let tangents at A and B meet at P(h,k) and include an angle 60.
If CPA=θ then CPB=θ+60
sinθ=2CP,sin(θ+60)=3CP
sin(θ+60)sinθ=32
or 2(12sinθ+32cosθ)=3sinθ
or 2sinθ=3cosθ or tanθ=32
But tanθ=CAPA=2S=2h2+k24
32=2h2+k24. Square and generalize.
3(x2+y24)=16
or 3(x2+y2)=28
It is again a concentric circle.
924344_1008208_ans_1656127202fb499aa541338292298c74.png

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