The locus of the point, the chord of contact of tangents from which to the circle $x^2+y^2=a^2$ subtends a right angle at the center is a circle of radius

Let$P (h, k)$ be the point. Then, the chord of contact of tangents drawn from $P$ to the circle

$x^2+y^2=a^2$ is $hx+ky=a^2$

The combined equation of the lines joining the (center) origin to the points of intersection of the circle

$x^2+y^2=a^2$ and the chord of contact of tangents drawn from $P(h, k)$ is a homogeneous equation of second degree given by

$x^2+y^2=a^2{\left(\frac{hx+ky}{a^2}\right)}^2\Rightarrow a^2\left(x^2+y^2\right)=(hx+ky{)}^2$

The lines given by the above equation will be perpendicular, if

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

$\Rightarrow h^2-a^2+k^2-a^2=0\Rightarrow h^2+k^2=2a^2$

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

Clearly, it is a circle of radius $\sqrt{2}a$