Hey guys,

I'm really bad at these types of questions, I don't know what it is about them but they always seem to stump me. Here it is,

The variable point $\displaystyle P(a\sec t, b\tan t)$ is on the hyperbola with equation $\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and N is the point (3a, 3b). The point Q lies on the line PN so that PQ = 2QN. As t varies find, in cartesian form, the locus of Q.

I tried to find the line PN by finding its gradient by

$\displaystyle \frac{3b - b\tan t}{3a - a\sec t}$

and then using the usual method for finding the equation of a straight line, that is

$\displaystyle \frac{y-3b}{x-3a} = \frac{3b - b\tan t}{3a - a\sec t}$

but then I get stuck and cannot see what to do next. Do I need to find the point Q? I know I need to eliminate t, but I just can't see how to approach this one.

Are there any general tips and tricks common to conic section questions like this? Given a formula I can identify which conic it is, and its directrix, foci etc. But when the questions get slightly harder (like this one) I just go to pieces.

Any help would be much appreciated :)

Stonehambey