Question

Find the equation of the hyperbola satisfying the given conditions:

Foci `(+-3sqrt5, 0)`, the latus rectum is of length 8.

Answer 1

Foci `(± 3sqrt5, 0)` the latus recum is of length 8.

Here, the foci are on the x-axis.

Therefore, the equation of the hyperbola is of the form `x^2/a^2 - y^2/b^2 = 1`

Since the foci are `(± 3sqrt5, 0)`, C = `±3sqrt5`

Length of latus retum = 8

`(2b^2)/a = 8`

= b² = 4a

We know that a² + b² = c²

∴ a² + 4a = 45

= a² + 4a - 45 = 0

= a² + 9a - 5a - 45 = 0

= (a + 9) (a - 5) = 0

= a = -9, 5

but a ≠ −9

Thus, the equation of the hyperbola is `x^2/25 - y^2/20 = 1`.