Question

Find the equation of the hyperbola satisfying the given conditions.
$\text{foci}(0,\pm \sqrt{10}),\text{passing through (2,3)}$

Answer 1

Here foci are $(0,\pm \sqrt{10})\text{which lie on y- axis}$
So the equation of hyperbola in standard form is $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1\therefore \text{foci}(0,\pm c)is(0,\pm \sqrt{10})\Rightarrow a=\sqrt{10}\text{We know that}{c}^2=a^2+b^2\therefore (\sqrt{10}{)}^2=a^2+b^2\Rightarrow b^2=10-a^2\text{Since the hyperbola passes through}(2,3)\therefore \frac{9}{a^2}-\frac{4}{10-a^2}=1\Rightarrow \frac{9\left(10{-}^2\right)-4a^2=a^2\left(10\right)-a^2}{a^2(10-a^2)}\Rightarrow a^4-23a^2+90=0\Rightarrow a^4-18a^2-5a^2+90=0\Rightarrow (a^2-18\left)\right(a^2-5)=0\Rightarrow a^2=18or{a}^2=5\text{When}{a}^2-18\text{then}{b}^2=10-18=-8\text{Which is not possible}\text{when}{a}^2=\text{then}{b}^2=10-5=5\text{Thus required equation of hyperbola is},\frac{y^2}{5}-\frac{x^2}{5}=1$