42nd Swedish 2002 Problem 3 C is the circle center (0,1), radius 1. P is the parabola y = ax 2 . They meet at (0, 0). For what values of a do they meet at another point or points?
a > ½ Solution The circle is x 2 + (y-1) 2 = 1, so if (x,y) lies on the parabola and the circle, then y 2 - 2y + y/a = 0, so y = 0 or 2-1/a. y = 0 corresponds to the origin, which always lies on both curves. y = 2-1/a can only lie on the circle if a ≥ 1/2. If a = 1/2, then we just get the origin again. So a > 1/2 is certainly a necessary condition for the curves to meet again. If a > 1/2, then we may put a = 1/2 + 2k2 and we find that the two points (±4k/(4k 2 +1), 8k 2 /(4k 2 +1)) lie on both curves. So the condition is also sufficient. 42nd Swedish 2002 © John Scholes jscholes@kalva.demon.co.uk 21 December 2003 Last corrected/updated 21 Dec 03