The angle between the pair of tangents drawn from a point P to the circle x2+y2+4x−6y+9sin2α+13cos2α=0 is 2α. Then the equation of the locus of the point P is
A
x2+y2+4x−6y+4=0
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B
x2+y2+4x−6y−9=0
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C
x2+y2+4x−6y−4=0
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D
x2+y2+4x−6y+9=0
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Solution
The correct option is Dx2+y2+4x−6y+9=0 The center of the circle x2+y2+4x−6y+9sin2α+13cos2α=0
is C(-2, 3) and its radius is √22+(−3)2−9sin2α−13cos2α⇒√4+9−9sin2α−13cos2α=|2sinα|
Let P(h, k) be any point on the locus. Then ∠APC=α
From the diagram. sinα=ACPC=2sinα√(h+2)2+(k−3)2or(h+2)2+(k−3)2=4orh2+k2+4h−6k+9=0
Thus, the required equation of the locus is x2+y2+4x−6y+9=0