[Math] Locus of two perpendicular tangents to a circle

Question

Describe the locus of the point of intersection of two perpendicular tangents to the circle $x^2+y^2=r^2$

I tried:
The circle center is $(0,0)$ and the radius $= r$ so would the point of intersection be $(\pm r,\pm r)$ ?

Answer 1

The two tangent points, their intersection and the center of the circle form the vertices of a square, so that the intersection is $\sqrt{2}r$ away from the center: the locus is the circle with the same center and radius $\sqrt{2}r$.