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Question

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



Let PA and PB be the tangents drawn from the point P(h, k) to the given circle with centre C(-2, 3). So, that
APB=2α and APC=CPB=αPAC=PBC=90From triangle PCA,sinα=CACP and CA=4+9(9sin2α+13cos2α)=2sinαCP=2.4=h2+k2+4h6k+13.
The locus of P(h, k) is
x2+y2+4x6y+9=0


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