Question

Find the equation of the hyperbola satisfying the give conditions: Foci $(0,\pm \sqrt{10})$
passing through $(2,3)$

Answer 1

Here the foci are on the $y$-axis.
Therefore, the equation of the hyperbola is of the form $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$
Since the foci are $(0,\pm \sqrt{10})\Rightarrow ae=c=\sqrt{10}$
We know that $a^2+b^2=c^2$
$\therefore$ $a^2+b^2=10$
$\Rightarrow b^2=10-a^2...\left(1\right)$
Since the hyperbola passes through point $(2,3)$
$\frac{9}{a^2}-\frac{4}{b^2}=1...\left(2\right)$
From equation (1) and (2), we obtain
$\frac{9}{a^2}-\frac{4}{(10-a^2)}=1$
$\Rightarrow 9(10-a^2)-4a^2=a^2(10-a^2)$
$\Rightarrow 90-9a^2-4a^2=10a^2-a^4$
$\Rightarrow a^4-23a^2+90=0$
$\Rightarrow$$a^4-18a^2-5a^2+90=0$
$\Rightarrow a^2(a^2-18)-5(a^2-18)=0$
$\Rightarrow$$(a^2-18)(a^2-5)=0$
$\Rightarrow$$a^2=18or5$
In hyperbola $c>a$, i.e., $c{}^2>a{}^2$
$\therefore a^2=5$
$\Rightarrow$$b^2=10-a^2=10-5=5$
Thus the equation of the hyperbola is $\frac{y^2}{5}-\frac{x^2}{5}=1$