### 11th Putnam 1951

**Problem A4**

Sketch the curve y^{4} - x^{4} - 96y^{2} + 100x^{2} = 0.

**Solution**

The graph is in three parts. It is symmetrical on reflection in the x-axis and on reflection in the y-axis. The central part is a figure of 8 with crossing point at the origin and touching the lines y/x = ±√(100/96). The top and bottom of the figure of 8 are at (0, ±√96). The right hand part is a sort of dimpled hyperbola with asymptotes y/x = ±1. It comes in along each asymptote and touches the line x = 8 at y = ±√48, then it dimples back away from the origin to cut the x-axis at x = 10. The left hand part of the curve is just the reflection of this in the y-axis.

11th Putnam 1951

© John Scholes

jscholes@kalva.demon.co.uk

5 Mar 2002