# Thread: Hyperbola Proof

1. ## Hyperbola Proof

Here's the question.

If P is any point on the hyperbola and PR and PS are drawn parallel to the asymptotes, the dPR x dPS = constant. Show this is true for the hyperbola:

x^2/4 - y^2/4 = 1

by choosing three different points for P. Can you prove the general case?

P.S. I'm assuming that dPR means the distance from P to R.

Here's the diagram

Thanks in advance.

2. ## Re: Hyperbola Proof

You could make the substitution

$x=\frac{\sqrt{2}}{2}(x'+y')$
$y=\frac{\sqrt{2}}{2}(-x'+y')$

which is equivalent to considering new axes Ox'y' turned 45 degrees clockwise. In the new coordinate system, the equation will be x'y' = 2 and PR and PS will simply be coordinates of P. You should also show that the transformation from (x, y) to (x', y') preserves distances, which is easy.