Find equation of hyperbola?

Question

The hyperbola has center $(0,0)$, and goes through the points $(3,1)$ and $(9,5)$ and the coordinate axes are the symmetry axes. The correct answer is $x^2 - 3y^2=6$.

Answer 1

The equation is $\frac{x^2}{a}-\frac{y^2}{b}=1$ for $a,b\neq 0$. Plugging in the two points you end up with two equations:

$9b-a=ab$

$81b-25a=ab$

So $a=\frac{9b}{b+1}$. This give you that $81b(b+1)-225b=9b^2$ so $b(72b-144)=0$. We obtain that $b=2$ and $a=6$ which give the answer you already know.