I don't understand how to do part (c). Can someone help? Also, if possible can someone draw it so I can see how it is pictured. Thanks in advance.
(c) Obviously M has coordinates $\displaystyle \left(\frac{cp + 2pc}{2}, \, \frac{\frac{c}{p} + \frac{c}{2p}}{2}\right)$, that is, $\displaystyle \left(\frac{3cp}{2}, \, \frac{3c}{4p}\right)$.
So the equation of the locus has parametric equations:
$\displaystyle x = \frac{3cp}{2}$ .... (1)
$\displaystyle y = \frac{3c}{4p}$ .... (2)
and $\displaystyle p \in R \setminus \{0\}$.
From (1): $\displaystyle p = \frac{2x}{3c}$.
From (2): $\displaystyle p = \frac{3c}{4y}$.
Therefore $\displaystyle \frac{2x}{3c} = \frac{3c}{4y} \Rightarrow xy = \frac{9 c^2}{8}$ ....