The angle between a pair of tangents from a point P to the circe `x^2 + y^2+ 4 x-6y + 9 sin2 alpha + 13 cos^2 alpha =0` is `2alpha`. Find the equation of the locus of the point P.

Options

A. `x^(2)+y^(2)+4x-6y+4=0`
B. `x^(2)+y^(2)+4x-6y-9=0`
C. `x^(2)+y^(2)+4x-6y-9=0`
D. `x^(2)+y^(2)+4x-6y+9=0`

Answer

D

Explanation

The equation of given circle is
`x^(2)+y^(2)+4x-6y+9sin ^(2)alpha+13cos ^(2)alpha=0`
`therefore` Centre`A(-2, 3)` in `DeltaABP,`
`sin alpha=(2 sin alpha)/(sqrt((h+2)^(2)(k-3)^(2)))`

`implies(h+2)^(2)+(k-3)^(2)=4`
`impliesh^(2)+k^(2)+4h-6k+9=0`
Hence, the required locus of `P (h, k)` is
`x^(2)+y^(2)+4x-=6y+9=0`