Find the equation of the hyperbola whose foci are and the length of the latus rectum is 8 units.

Given: Foci are the length of the latus rectum is 8 units

Need to find: The equation of the hyperbola.

Let, the equation of the hyperbola be:

The length of the latus rectum is 8 units.

Therefore, ⇒ ---- (1)

The foci are given at

That means, ae =, where e is the eccentricity.

We know that,

Therefore,

⇒

⇒

⇒ [Squaring both sides]

⇒ [From (1)]

⇒

⇒

⇒

So, either a = 5 or, a = -9

That means, either b = or, b =

The value of b = is not a valid one. So, the b value and its corresponding a value is not acceptable.

Hence, the acceptable value of a is 5 and b is

So, the equation of the hyperbola is,

⇒ [Answer]