Question

Find the equation of the hyperbola satisfying the given conditions:

Foci `(0, +- sqrt10)`, passing through (2, 3)

Answer 1

Foci `(0, ±sqrt10)`

⇒ The transverse axis is along the y-axis.

and c = `sqrt10` or c² = 10 = a² + b²

∴ a² + b² = 10     ……(i)

Let the equation of hyperbola

`y^2/a^2 - x^2/b^2 = 1`

It goes from the point (2, 3)

∴ `9/a^2 - 4/b^2 = 1` or 9b² − 4a² = a²b²

By substituting the value of b² from equation (i)

= 9(10 − a²) − 4a² = a² (10 − a²)

= 90 − 9a² − 4a² = 10a² − a⁴

= a⁴ − 23a² + 90 = 0

= a⁴ - 18a² - 5a² + 90 = 0

= (a² − 18)(a² − 5) = 0 

= a² = 18 or 5

When, a² = 18, b² = 10 − a²

= 10 − 18

= −8

Hence, a² ≠ 18

When a² = 5, b² = 10 − 5 = 5

∴ equation of hyperbola

`y^2/a^2 - x^2/b^2 = 1`

or `y^2/5 - x^2/5 = 1`