# Thread: Hyperbola equation solving

1. ## Hyperbola equation solving

Prove that the stright lines x/a - y/b = m and x/a + y/b = 1/m where a and b are given positive real numbers and m is a parameter always meet on a hyperbola.

No idea how to go about this one. Does this mean that the lines are perpendicular to each other?

2. ## Re: Hyperbola equation solving

Originally Posted by Don
Prove that the stright lines x/a - y/b = m and x/a + y/b = 1/m where a and b are given positive real numbers and m is a parameter always meet on a hyperbola.

No idea how to go about this one. Does this mean that the lines are perpendicular to each other?
1. Eliminate the parameter m: Plug in the LHS of the 1st equation into the 2nd equation:

$\frac xa + \frac yb = \frac1{\frac xa - \frac yb}$

2. Multiply both sides by $\left( \frac xa - \frac yb \right)$ . You'll get

$\displaystyle{\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1}$

which is the equation of a hyperpola with it's center at the origin.

Thanks.