MathDB

4

Part of 1973 IMO Longlists

Problems(1)

Find the locus of the center of the band

Source: IMO LongList 1973 - P4

6/6/2011
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 PP, such that the two tangents from PP to the ellipse with equation x2a2+y2b2=1\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 x2+y2=a2+b2x^2 + y^2 = a^2 + b^2.
conicsellipsegeometry unsolvedgeometry