[SOLVED] Hyperbola 3

**UltraGirl** · April 16th 2010, 05:06 PM

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?

**Grandad** · April 17th 2010, 10:23 AM

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 $P\;\Big(cp,\frac cp\Big)$ 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?

Grandad

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