How can I find the impact parameter in hyperbolic geometry?

[ColdFusion85]

Homework Statement

Prove that the magnitude of the impact parameter B equals the length (-b) of the hyperbolic semiminor axis.

Homework Equations

|B|=|b|=|a|sqrt(e^2-1)

The Attempt at a Solution

I really don't know where to start. I was thinking of finding a relation between a and b but that is just the right hand side of the equation above (for hyperbolic geometry). I was thinking about the Pythagorean theorem, but then I don't know how |B|=|a+b| would become |a|sqrt(e^2-1). Could anyone help me along the right direction? Thanks.

 

[ColdFusion85]

Anyone?

 

[ColdFusion85]

bumping one more time, is there anyone out there that can help?

 

[Dick]

I would do it this way. Take a general hyperbola in standard position. x^2/a^2-y^2/b^2=1. That puts a vertex at the point xy point (sqrt(a^2+b^2),0). An asymptote is the line y=bx/a. The impact parameter is the distance from the vertex to the asymptote line. If you work that out you should get b. Which is the semi-major axis.