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Question

# The angle between the pair of tangents drawn from a point P to the circle x2+y2+4x−6y+9sin2α+13 cos2α=0 is 2α. Then 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+4x−6y+9=0The center of the circle x2+y2+4x−6y+9 sin2α+13 cos2α=0 is C(-2, 3) and its radius is √22+(−3)2−9 sin2α−13 cos2α⇒√4+9−9 sin2α−13 cos2α=|2 sin α| Let P(h, k) be any point on the locus. Then ∠APC=α From the diagram. sin α=ACPC=2 sin α√(h+2)2+(k−3)2or (h+2)2+(k−3)2=4or h2+k2+4h−6k+9=0 Thus, the required equation of the locus is x2+y2+4x−6y+9=0

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