[maths]P(cp,\frac{c}{p})[/maths]
[maths]Q(cq,\frac{c}{q})\Rightarrow (\frac{cp}{2},\frac{2c}{p})[/maths]
The coordinates of midpoint M is given by:
[maths]x=\frac{cp+\frac{cp}{2}}{2}=\frac{3cp}{4}[/maths]
[maths]y=\frac{\frac{c}{p}+\frac{2c}{p}}{2}=\frac{3c}{2p}[/maths]
[maths]xy=(\frac{3cp}{4})(\frac{3c}{2p})=\frac{9c^2}{8}=d^2[/maths] which is in form of parabola
For the other question, use the fact that angle in semi-circle=90 degrees by using gradient formula