In each of the following find the equations of the hyperbola satisfying the given conditions

foci , the latus-rectum = 8

Given: Foci and the latus-rectum = 8

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 (±ae, 0)

Length of latus rectum is

According to the question:

We know,

b² = a²(e² – 1)

⇒ 4a = 45 – a²

⇒ a² + 4a – 45 = 0

⇒ a² + 9a – 5a – 45 = 0

⇒ a(a + 9) – 5(a + 9) = 0

⇒ (a + 9)(a – 5) = 0

⇒ a = -9 or a = 5

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

⇒ a = 5

⇒ a² = 25

b² = 4a

⇒ b² = 4(5)

⇒ b² = 20

Hence, equation of hyperbola is: