Foci ( 4, 0), the latus rectum is of length 12

Here, the foci are on x-axis.

Thus,

The equation of the hyperbola is of the form

Since, the foci are ( 4, 0), c = 4

Length of latus rectum = 12

⇒

⇒ b² = 6a

We know that, a² + b² = c²

a² + 6a = 16

⇒ a² + 6a -16 = 0

⇒ a² + 8a – 2a - 16 = 0

⇒ (a +8)(a -2) = 0

⇒ A = -8, 2

Since, a is non – negative, a = 2

Thus, b² = 6a = 6 × 2 = 12

Hence, the equation of the hyperbola is