Find the equation of the hyperbola satisfying the given conditions: Foci (0, ± √10), passing through (2, 3)

Solution:

Foci (0, ± √10), passing through (2, 3).

Here, the foci are on the y-axis.

Therefore,

the equation of the hyperbola is of the form y²/a² - x²/b² = 1

Since the Foci are (0, ± √10), c = √10

We know that, c² = a² + b²

Hence,

⇒ a² + b² = 10

⇒ b² = 10 - a² ....(1)

Since the hyperbola passes through point (2, 3)

9/a² - 4/b² = 1 ....(2)

From equations (1) and (2), we obtain

9/a² - 4/(10 - a²) = 1

⇒ 9 (10 - a²) - 4a² = a² (10 - a²)

⇒ 90 - 9a² - 4a² = 10a² - a⁴

⇒ a⁴ - 23a² + 90 = 0

⇒ a⁴ - 18a² - 5a² + 90 = 0

⇒ a² (a² - 18) - 5(a² - 18) = 0

⇒ (a² - 18)(a² - 5) = 0

⇒ a² = 18 or 5

In hyperbola, c > a , i.e., c² > a²

Therefore,

⇒ a² = 5

⇒ b² = 10 - a²

⇒ b² = 10 - 5

⇒ b² = 5

Thus, the equation of the hyperbola is y²/5 - x²/5 = 1

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NCERT Solutions Class 11 Maths Chapter 11 Exercise 11.4 Question 15

Find the equation of the hyperbola satisfying the given conditions: Foci (0, ± √10), passing through (2, 3)

Summary:

The hyperbola equation is y²/5 - x²/5 = 1 while the foci (0, ± √10) pass-through (2, 3)