35th Putnam 1974 Problem A5 The parabola y = x 2 rolls around the fixed parabola y = -x 2 . Find the locus of its focus (initially at x = 0, y = 1/4). Solution
: horizontal line y = 1/4. The tangent to the fixed parabola at (t, -t 2 ) has gradient -2t, so its equation is (y + t 2 ) = -2t(x - t). We wish to find the reflection of the focus of the fixed parabola in this line, because that is the locus of the rolling parabola when they touch at (t, -t 2 ). The line through the focus of the fixed parabola (0, -1/4) and its mirror image is perpendicular to the tangent and so has gradient 1/(2t). Hence its equation is (y + 1/4) = x/(2t). Hence it intersects the tangent at the point with y + t 2 = -4t 2 (y + 1/4) + 2t 2 or y(1 + 4t 2 ) = 0, hence at the point (t/2, 0). So the mirror image is the point (t, 1/4). t can vary from -∞ to +∞, so the locus is the horizontal line through (0, 1/4). 35th Putnam 1974 © John Scholes jscholes@kalva.demon.co.uk 18 Aug 2001