### 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**

Answer: 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