Distance from point to line on hyperbola asymptotes problem

Hi,

I've been struggling with this for a while now and to no avail. Here's the question.

A hyperbola of the form

has asymptotes with equations and passes through the point (a,0). Find the equation of the hyperbola in terms of x, y, a and m.

A point P on this hyperbola is equidistant from one of its asymptotes and the x-axis. Prove that, for all values of m, P lies on the curve with equation.

I hate these type of questions as I always end up with a pile of messy algebra and ton of screwed up paper. Here's my attempt this time.

I got the first part ok, the equation of the hyperbola is

For the second part I decided to pick an arbitrary point on the hyperbola and find its distance from the asymptote and the x-axis and set them equal, then eliminate m. So my general point is . Now the distance of this point to the x-axis is simply . For the moment I chose the point to be in the first quadrant so the equation of the corresponding asymptote is . Wolfram says that the distance between the point and the asymptote is

where the line is

So for my point and line this becomes

which I set equal to the distance from the y-axis to obtain

It is here where I have no idea how to eliminate m to get the required equation or even if I've been doing it wrong the whole time.

Please can someone help, thanks!!

Stonehambey