# [SOLVED] Hyperbola 3

• Apr 16th 2010, 05:06 PM
UltraGirl
[SOLVED] Hyperbola 3
Hello.

Prove that the locus of midpoints of parallel chords of xy = c^2 is a diameter.

Is anyone able to give me some hints on how to do this question?
• Apr 17th 2010, 10:23 AM
Hello UltraGirl
Quote:

Originally Posted by UltraGirl
Hello.

Prove that the locus of midpoints of parallel chords of xy = c^2 is a diameter.

Is anyone able to give me some hints on how to do this question?

Consider the chord joining $P\;\Big(cp,\frac cp\Big)$ to $Q\;\Big(cq,\frac cq\Big)$.

You can easily show that its gradient is $-\frac1{pq}$.

A set of parallel chords will all have the same gradient, $m$, say. So:
$-\frac1{pq}=m$, where $m$ is a constant. Call this equation (1).
The mid-point $(x,y)$ of $PQ$ satisfies:
$x = \tfrac12c(p+q)$
and
$y = \tfrac12(\frac cp+\frac cq\Big)$
$=\tfrac12c\Big(\frac{p+q}{pq}\Big)$
Now, using equation (1), find an equation connecting $x, y$ and $m$, and show that this, the locus of the mid-point of $PQ$, is a straight line through the origin.

Can you complete it now?