Question

Find the equation of the hyperbola whose foci are `(0,+- sqrt10)` and passing through the point (2,3)

Answer 1

`foci -> (0, +- sqrt10)`

be = `sqrt10` ......(1)

Equation of hyperbola is

`y^2/b^2 - x^2/a^2 = 1`  .......(2)

equation 2 passing through (2, 3)

`:. 9/b^2 - 4/a^2 = 1`

we know that `a^2 = b^2e^2 - b^2`

`a^2 = 10 -  b^2`

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

Put `b^2 = t`

`9/t- 4/(10 - t) = 1`

`90 - 9t - 4t = 10t - t^2`

`t^2 - 23t  + 90 = 0`

t = 18, t = 5

`b^2 = 18, b^2 = 5`

`b = 3sqrt2 , b =  sqrt5`

When `b = 3sqrt2`

then `a^2 = 10 - 18 = -8`   (Neglected `∵ a = sqrt-8` is an imaginary number).

∴ when `b =  sqrt5`

then `a =  sqrt5`

∴ equation of hyperbola

`y^2/5 - x^2/5  = 1`

`y^2 - x^2 = 5`