# Thread: [SOLVED] Hyperbola 3

1. ## [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?

2. Hello UltraGirl
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?