Question

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

Answer 1

Foci (0, $±\sqrt{10}$) and passing through (2, 3)
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$(±\sqrt{10}, 0), c = \sqrt{10}$

We know that $a^2 \+ b^2 = c^2$
$∴ b^2 = 10 – a^2 ………….. (1)$

Since the hyperbola passes through point (2, 3),

$\frac{9}{a^2} – \frac{4}{b^2} = 1 … (2)$
From equations (1) and (2), we obtain

$\frac{9}{a^2} – \frac{4}{(10-a^2)} = 1$

$⇒ 9(10 – a^2) – 4a^2 = a^2(10 –a^2)$

$⇒ 90 – 9a^2 – 4a^2 = 10a^2 – a^4$

$⇒ a^4 – 23a^2 \+ 90 = 0$

$⇒ a^4 – 18a^2 – 5a^2 \+ 90 = 0$

$⇒ a^2(a^2 -18) -5(a^2 -18) = 0$

$⇒ (a^2 – 18) (a^2 -5) = 0$

$⇒ a^2 = 18$ or $5$

In hyperbola, $c > a, i.e., c^2 > a^2$

$∴ a^2 = 5$
$⇒ b^2 = 10 – a^2= 10 – 5= 5$

Thus, the equation of the hyperbola is $\frac{y^2}{5} – \frac{x^2}{5} = 1$