Find the equation of hyperbola which has Foci (0, 10 ) and passin…

Question

Find the equation of hyperbola which has Foci $(0, \pm \sqrt {10} )$ and passing through $(2, 3)$

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Answer

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

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