# Math Help - Rectangular Hyperbola

1. ## Rectangular Hyperbola

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.

2. Originally Posted by Air
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 $\left(\frac{cp + 2pc}{2}, \, \frac{\frac{c}{p} + \frac{c}{2p}}{2}\right)$, that is, $\left(\frac{3cp}{2}, \, \frac{3c}{4p}\right)$.

So the equation of the locus has parametric equations:

$x = \frac{3cp}{2}$ .... (1)

$y = \frac{3c}{4p}$ .... (2)

and $p \in R \setminus \{0\}$.

From (1): $p = \frac{2x}{3c}$.

From (2): $p = \frac{3c}{4y}$.

Therefore $\frac{2x}{3c} = \frac{3c}{4y} \Rightarrow xy = \frac{9 c^2}{8}$ ....