# Locus of two perpendicular tangents to a circle

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 $(\pm r,\pm r)$ ?

• The tangents are not necessarily parallel to the axes. Commented Jun 23, 2017 at 18:16

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 $\sqrt{2}r$.

the first tangent is at the point $(r\cos (t),r\sin (t))$ of the circle.

the second tangent is perpendicular and its point is

$$r\cos (t+\pi/2),r\sin (t+\pi/2))=(-r\sin (t),r\cos (t))$$

the two lines intersection point is

$$(r\sqrt {2}\cos (t+\pi/4),r\sqrt {2}\sin (t+\pi/4)) .$$

• Don't you missed a $\sqrt{2}$? Commented Jun 23, 2017 at 18:38
• @enzotib Yes yes thanks .i edited. Commented Jun 23, 2017 at 19:09