Question 2
A circle has two parallel chords $AB$ and $CD$. A point $M$ moves on the circle, and the line $MD$ meets the line $AB$ at $Q$.
(a) As $M$ tends to $D$ or to $C$, where does the circumcenter of triangle $MCQ$ tend? Determine the locus of the center of $(MCQ)$.
(b) Take a fixed point $E$ outside the plane of the diagram. How should $E$ be chosen so that the locus of the center of the sphere $(MCQE)$ coincides with the locus of the center of $(MCQ)$?
(a) As $M$ tends to $D$ or to $C$, where does the circumcenter of triangle $MCQ$ tend? Determine the locus of the center of $(MCQ)$.
(b) Take a fixed point $E$ outside the plane of the diagram. How should $E$ be chosen so that the locus of the center of the sphere $(MCQE)$ coincides with the locus of the center of $(MCQ)$?