### Putnam 1994

**Problem A2**

Let R be the region in the first quadrant bounded by the x-axis, the line 2y = x, and the ellipse x^{2}/9 + y^{2} = 1. Let R' be the region in the first quadrant bounded by the y-axis, the line y = mx, and the ellipse. Find m such that R and R' have the same area.

**Solution**

*Easy.*

Stretch the figure by a factor 3 along the y-axis, so that the point (x, y) goes to (x, 3y). Then the ellipse becomes a circle and the line 2y = x, becomes the line 2y = 3x. Obviously, the required line in the stretched figure is 3y = 2x. Shrinking back to the original preserves the ratio of the areas and gives m = 2/9.

Putnam 1994

© John Scholes

jscholes@kalva.demon.co.uk

12 Dec 1998