Question

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

Answer 1

let the coordinates of P are`(h,k)`
the angle made by P to the tangents is `2alpha`
the centre of the circle is C`(-2,3)`
so, `r= sqrt(4+9-9sin alpha - 13cos^2 alpha)`
`r= sqrt(4+9cos^2 alpha-13cos^ alpha) `
`= sqrt(4-4cos^2 alpha)`
`=sqrt(4(sin^2 alpha))`
`=2sin alpha`
in `/_ APC`
`sin alpha= (AC)/(PC)= (2sin alpha)/(sqrt((h+2)^2 + (k-3)^2)`
`= sqrt((h+2)^2 + (k-3)^2) = 2`
`h^2 + k^2 +4k-6k + 13=4`
locus of P(h,k) is `x^2+y^2+4x-6y+9=0`
option D is correct