# [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 $\displaystyle P\;\Big(cp,\frac cp\Big)$ to $\displaystyle Q\;\Big(cq,\frac cq\Big)$.

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

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

Can you complete it now?