ISL 1973 Problem 4

A circle of radius 1 is placed in a corner of a room (i.e., it touches the horizontal floor and two vertical walls perpendicular to each other). Find the locus of the center of the band for all of its possible positions.Note. For the solution of this problem, it is useful to know the following Monge theorem: The locus of all points $P$ , such that the two tangents from $P$ to the ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are perpendicular to each other, is a circle - a so-called Monge circle - with equation $x^2 + y^2 = a^2 + b^2$ .