Given: Foci (±4, 0), the latus-rectum = 12

To find: equation of the hyperbola

Formula used:

Standard form of the equation of hyperbola is,

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

Length of latus rectum is

According to the question:

ae = 4

We know,

b² = a²(e² – 1)

⇒ 6a = 16 – a²

⇒ a² + 6a – 16 = 0

⇒ a² + 8a – 2a – 16 = 0

⇒ a(a + 8) – 2(a + 8) = 0

⇒ (a + 8)(a – 2) = 0

⇒ a = -8 or a = 2

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

⇒ a = 2

⇒ a² = 4

b² = 6a

⇒ b² = 6(2)

⇒ b² = 12

Hence, equation of hyperbola is: