Question

If the distance between the foci of a hyperbola is 16 and its eccentricity is `sqrt(2)`, then obtain the equation of the hyperbola.

Answer 1

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

Distance between the foci = 2ae

2ae = 16

⇒ ae = 8

⇒ `a xx sqrt(2)` = 8

⇒ `a = 8/sqrt(2) = 4sqrt(2)`  ......`[∵ e = sqrt(2)]`

Now `b^2 = a^2(e^2 - 1)`

⇒ `b^2 - (4sqrt(2))^2 (2 - 1)`

⇒ b² = 32

a = `4sqrt(2)`

⇒ a² = 32

⇒ x² – y² = 32

Hence, the required equation is `x^2/32 - y^2/32` = 1