Let $(C)$ be a circle of center $O$ and radius $r$. Let $E$ be any point on $(C)$. Let $P$ be any point other than $E$ in the plane. The perpendicular bisector of $[PE]$ cuts $(OE)$ in a point $M$. The objective is:
Find the locus of the point $M$ as $P$ varies.
The wording of the problem is quite confusing. If $E$ is assumed to be fixed, then both $O$ and $E$ are fixed, so can't we say that the locus of $M$ is the straight line $(OE)$? (or possibly some section of this line). So I'm thinking that the following wording is the one intended:
Find the locus of the point $M$ as $P$ and $E$ vary.
This is a problem in a section titled $"Conics"$ in some high school Math test. I've tried to find a certain (conic) pattern in $M$'s movements, but I couldn't figure out one. There's a hint on the side: "Discuss according to the different positions of $P$ with respect to $(C)$".
I think that the problem is missing some additional details. So I'll ask two questions:
1) In case the problem is missing some extra details, what are the slight touches that we may add to it so that it becomes a good one (and related to conic sections)?
2) In case nothing is missing, what is the solution?
Thanks a lot.
Yes you are right. A better problem would be, The locus of M as E varies keeping everything else fixed, in which the locus comes out to be a hyperbola if P is outside the circle, ellipse if inside and circle if on the center O.