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?

Printable View

- Apr 16th 2010, 05:06 PMUltraGirl[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 AMGrandad
Hello UltraGirlConsider 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

$\displaystyle y = \tfrac12(\frac cp+\frac cq\Big)$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.

$\displaystyle =\tfrac12c\Big(\frac{p+q}{pq}\Big)$

Can you complete it now?

Grandad