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Question

The angle between the pair of tangents drawn from a point P to the curve x2+y2+4x6y+9sin2α+13cos2α=0 is 2α. The locus of P is

A
x2+y2+4x6y+4=0
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B
x2+y2+4x6y9=0
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C
x2+y2+4x6y4=0
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D
x2+y2+4x6y+9=0
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Solution

The correct option is D x2+y2+4x6y+9=0
Center of the circle
x2+y2+4x6y+9sin2α+13cos2α=0
is C(2,3) and its radius is
(2)2+(3)29sin2α13cos2α=1313cos2α92sin2α=13sin2α9sin2α=4sin2α=2sinα
Let (h,k) be any point P and APC=α,PAC=π2
That is, triangle APC is a right angle triangle
sinα=ACPC=2sinα(h+2)2+(k3)2(h+2)2+(k3)2=4h2+4+4h+k2+96k=4h2+y2+4x6y+9=0
Thus, required equation of locus is
x2+y2+4x6y+9=0
348117_135261_ans.PNG

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