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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 is2α. The equation of the locus of the point P is


Your Answer
A
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B
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C
Correct Answer
D

Solution

The correct option is D
Let P(x1,y1) be a point in the locus
Radius of Sx2+y2+4x6y+9sin2α+13cos2α=0
r=(2)2+(3)29sin2α13cos2α=13(1cos2α)9sin2α=2sinα
Length of the tangent from P(x1,y1) to S = 0 is S11=x21+y21+4x16y1+9sin2α+13cos2α
Since 2α is the angle between the tangent drawn from P to S = 0
we have tanα=rs11S11.tan2α=r2(x21+y21+4x16y1+9sin2α+13cos2α)tan2α=4sin2α(x21+y21+4x16y1+9+4cos2α)sin2αcos2α=4sin2α(x21+y21+4x16y1+9)sec2α=0x21+y21+4x16y1+9=0Locus of (x1,y1) is x2+y2+4x6y+9=0

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