Two perpendicular tangents to the circle $x^2+y^{2 }=a^2$ meet at $P.$ Then the locus of $P$has the equation

Let the coordinates of $P$ be $(h, k)$. Then, the equation of the tangents drawn from $P(h,k)$ to $x^2+y^2=a^2$ is 

$\left(x^2+y^2-a^2\right)\left(h^2+k^2-a^2\right)={\left(hx+ky-a^2\right)}^2$                  [Using $SS\text{'} =T^{2 }$]

This equation represents a pair of perpendicular lines. 

$\text{ Coeff. of }{x}^2+\text{ Coeff. of }{y}^2=0$ 

$\Rightarrow \left(h^2+k^2-a^2-h^2\right)+\left(h^2+k^2-a^2-k^2\right)=0$ 

$\Rightarrow h^2+k^2=2a^2$ 

Hence, the locus of  $(h,k)$is $x^2+y^2=2a^2.$