Originally Posted by
earboth Here comes a slightly different approach:
1. Let P denotes an arbitrary point on the hyperbola, r the distance of P to the focus F and l the distance of P to the directrix. Then
$\displaystyle \dfrac rl = e~\implies~r = e \cdot l$
2. Plug in the values you know:
$\displaystyle \sqrt{(x-1)^2+(y+3)^2} = 1.5(y+2)$
3. Simplify and re-arrange so that you'll get the equation of the hyperbola in standard form. I've got:
$\displaystyle -\dfrac{(x-1)^2}{\frac95}+\dfrac{\left(y+\frac65\right)^2}{\l eft(\frac56\right)^2}} = 1$