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

Solution:

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

Since the length of latus rectum is 12,

Then,

⇒ 2b²/a = 12

⇒ b² = 6a

We know that,

c² = a² + b²

Hence,

⇒ a² + 6a = 16

⇒ a² + 6a - 16 = 0

⇒ a² + 8a - 2a - 16 = 0

⇒ a (a + 8) - 2 (a + 8) = 0

⇒ (a + 8)(a - 2) = 0

⇒ a = - 8, 2

Since a is non-negative, a = 2

Therefore,

⇒ b² = 6a

⇒ b² = 6 x 2

⇒ b² = 12

Thus, the equation of the hyperbola is x²/4 - y²/12 = 1

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NCERT Solutions Class 11 Maths Chapter 11 Exercise 11.4 Question 13

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

Summary:

The equation of the hyperbola is x²/4 - y²/12 = 1 while the foci are (± 4, 0) and the latus rectum is of length 12