Find the equation of the hyperbola with foci `(0, +- sqrt(10))`, passing through (2, 3)

#### Solution

Given that: foci `(0, +- sqrt(10))`

∴ ae = `sqrt(10)`

⇒ `a^2e^2` = 10

We know that `b^2 = a^2(e^2 - 1)`

⇒ `b^2 = a^2e^2 - a^2`

⇒ `b^2 = 10 - a^2`

Equation of hyperbola is `y^2/a^2 - x^2/b^2` = 1

⇒ `y^2/a^2 - x^2/(10 - a^2)` = 1

If it passes through the point (2, 3) then

`9/a^2 - 4/(10 - a^2)` = 1

⇒ `(90 - 9a^2 - 4a^2)/(a^2(10 - a^2))` = 1

⇒ 90 – 13a^{2} = a^{2}(10 – a^{2})

⇒ 90 – 13a^{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, a^{2} = 5

∴ b^{2} = 10 –18 = – 8 and b^{2} = 10 – 5 = 5

b ≠ – 8

∴ b^{2} = 5

Here, the required equation is `y^2/5 - x^2/5` = 1 or y^{2} – x^{2} = 5.