Given: Foci (0, ±12), the latus-rectum = 36

To find: equation of the hyperbola

Formula used:

The standard form of the equation of the hyperbola is,

Coordinates of the foci for a standard hyperbola is given by (0, ±be)

Length of latus rectum is

According to the question:

be = 12

We know,

a² = b²(e² – 1)

⇒ 18b = 144 – b²

⇒ b² + 18b – 144 = 0

⇒ b² + 24b – 6b – 144 = 0

⇒ b(b + 24) – 6(b + 24) = 0

⇒ (b + 24)(b – 6) = 0

⇒ b = -24 or b = 6

Since b is a distance, and it can’t be negative

⇒ b = 6

⇒ b² = 36

a² = 18b

⇒ a² = 18(6)

⇒ b² = 108

Hence, equation of hyperbola is: