M is the midpoint of XY. The points P and Q lie on a line through Y on opposite sides of Y, such that |XQ| = 2|MP| and |XY|/2 < |MP| < 3|XY|/2. For what value of |PY|/|QY| is |PQ| a minimum?
Solution
Let the angle between the line through Y and XY be θ. Take Y' on the line such that XY' = XY. If P is on the opposite side of Y to Y', then Q is on the opposite side of Y' to Y. As P approaches Y, Q approaches Y' so the minimum value of PQ is YY', corresponding to PY/QY = 0. But it is unrealised, since the problem requires MY < MP. If P is on the same side of Y as Y', then as P approaches the midpoint of YY', Q approaches Y. So the minimum value of PQ is YY'/2 with PY/QY = infinity. Again it is unrealised because the problem requires MY < MP. P is allowed to be on either side of Y, so the unrealised minimum value of PQ is YY'/2 as PY/QY approaches infinity.
Comment. This seems too easy. Also the condition MP < 3XY/2 is not used. Is the question correctly stated?
© John Scholes
jscholes@kalva.demon.co.uk
26 Aug 2002