Problem:
Find the equation for the hyperbola which has foci $$F_1 = (-1, 3)$$ $$F_2 = (3,3) $$ and eccentricity $$\varepsilon = 2$$
Hint: Use a translation which moves the foci to the x-axis.
My attempt:
Using a simple translation $$\textbf{R} = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & -3 \\ 0 & 0 & 1\end{bmatrix}$$ I have translated the hyperbola 3 units down, such that the foci are on the x-axis.
I am not able to progress from here, and I can't find any formulae to help me.
As $y=3$ contains the foci, it also contains the major axis
If the equation is $\dfrac{(x-\alpha)^2}{a^2}-\dfrac{(y-\beta)^2}{b^2}=1$
As the center is the midpoint of the foci, we have $2\alpha=-1+3,2\beta=3+3$
Now the coordinates of the foci are $(\alpha\pm a\varepsilon,\beta)$
So, $1+2a=3,1-2a=-1\implies a=1$
We know $b^2=a^2(\varepsilon^2-1)$
Hope you can take it form here