S is a solid square side 2a. It lies in the quadrant x ≥ 0, y ≥ 0, and it is free to move around provided a vertex remains on the x-axis and an adjacent vertex on the y-axis. P is a point of S. Show that the locus of P is part of a conic. For what P does the locus degenerate?
Let A be the vertex that moves along the x-axis and B the vertex that moves along the y-axis. Suppose that when AB is horizontal P has coordinates b, c. In the general configuration let be the angle BAO be θ. Then P has coordinates x = (2a - b) cos θ + c sin θ, y = b sin θ + c cos θ. Hence cx - (2a - b)y = (b2 + c2 - 2ab) sin θ, bx - cy = (2ab - b2 - c2) cos θ. Squaring and adding we eliminate θ to get: (b2 + c2) x2 - 4ac xy + (4a2 + b2 + c2 - 4ab) y2 = (b2 + c2 - 2ab)2, which is the equation of a conic. So the locus of P must form part of this conic.
The conic degenerates if b2 + c2 = 2ab. In this case, the equation becomes 2ab x2 - 4ac xy + (4a2 - 2ab)y2 = 0, or bx2 - 2c xy + (2a - b)y2 = 0, or b2x2 - 2bc xy + c2 y2 = 0, or bx = cy. So in this case the locus lies on a straight line. We may write the condition b2 + c2 = 2ab as (a - b)2 + c2 = a2, which shows that such P lie on the semicircle diameter AB.
5th Putnam 1942
© John Scholes
5 Mar 2002