Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

The equation of given circle is $x^2+y^2+4x–6y+9si{n}^2\alpha +13co{s}^2\alpha =0$ Since tangents subtends an angle $2\alpha$ .
$\Rightarrow sin\alpha =\frac{2sin\alpha }{\sqrt{(h+2{)}^2+(k-3{)}^2}}\Rightarrow (h+2{)}^2+(k-3{)}^2=4\Rightarrow h^2+k^2+4h-6k+9=0\text{Hence, the required locus of}P(h,k)is{x}^2+y^2+4x-6y+9=0$