What is the locus of the point (in the plane), the sum of whose distances to a given point and line is fixed?
Solution
Take the line to be L, the point O and the distance sum k. Consider first points P on the same of L as O. Take a line L' parallel to L on the same side and a distance k from L. P meets the condition iff it is a distance k-d from L and hence d from L'. In other words P must be equidistant from O and L'. Hence it must lie on the parabola with focus O and directrix L'. However, also lies between L and L', so the locus is the part of the parabola between the two lines.
Similarly, if P lies on the other side of L' then it must be on the parabola focus O and directrix L", where L" is also parallel to L and a distance k from it. The complete locus is a sort of irregular oval.
© John Scholes
jscholes@kalva.demon.co.uk
1 Nov 2003
Last corrected/updated 1 Nov 03