2nd CaMO 1970 Problem 8 P lies on the line y = x and Q lies on the line y = 2x. Find the equation for the locus for the midpoint of PQ, if |PQ| = 4.
Let P be (a, a), Q be (b, 2b). Then the midpoint is ( (a+b)/2, (a+2b)/2) = (x, y). We have (a - b) 2 + (a - 2b) 2 = 16, so 2a 2 - 6ab + 5b 2 = 16. We want to recast this as an equation connecting x and y. We have a = 2(2x - y), b = 2(y - x), so 8(2x - y) 2 - 24(2x - y)(y - x) + 20(y - x) 2 = 15, or 2(4x 2 - 4xy + y 2 ) + 6(2x 2 - 3xy + y 2 ) + 5(x 2 - 2xy + y 2 ) = 4. So 25x 2 -36xy + 13y 2 = 4. The locus is a long, thin ellipse centered on the origin. 2nd CaMO 1970 © John Scholes jscholes@kalva.demon.co.uk 10 June 2002