The parabola y = x2 rolls around the fixed parabola y = -x2. Find the locus of its focus (initially at x = 0, y = 1/4).
Answer: horizontal line y = 1/4.
The tangent to the fixed parabola at (t, -t2) has gradient -2t, so its equation is (y + t2) = -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, -t2). 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 + t2 = -4t2(y + 1/4) + 2t2 or y(1 + 4t2) = 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
18 Aug 2001