Question

Find the equation of the hyperbola satisfying the give conditions: Foci (±4, 0), the latus rectum is of length 12

Answer 1

Foci (±4, 0), the latus rectum is of length 12.
Here, the foci are on the x-axis.
Therefore, the equation of the hyperbola is of the form X²/a² - Y²/b² =1.
Since the foci are (±4, 0), c = 4.
Length of latus rectum = 12

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.
∴b² = 6a = 6 × 2 = 12

Thus, the equation of the hyperbola is X²/4 - Y²/12 =1.