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# 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 y^{2}/a^{2} - x^{2}/b^{2} = 1

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

We know that, c^{2} = a^{2} + b^{2}

Hence,

⇒ a^{2} + b^{2} = 10

⇒ b^{2} = 10 - a^{2} ....(1)

Since the hyperbola passes through point (2, 3)

9/a^{2} - 4/b^{2} = 1 ....(2)

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

9/a^{2} - 4/(10 - a^{2}) = 1

⇒ 9 (10 - a^{2}) - 4a^{2} = a^{2} (10 - a^{2})

⇒ 90 - 9a^{2} - 4a^{2} = 10a^{2} - a^{4}

⇒ a^{4} - 23a^{2} + 90 = 0

⇒ a^{4} - 18a^{2} - 5a^{2} + 90 = 0

⇒ a^{2} (a^{2} - 18) - 5(a^{2} - 18) = 0

⇒ (a^{2} - 18)(a^{2} - 5) = 0

⇒ a^{2} = 18 or 5

In hyperbola, c > a , i.e., c^{2} > a^{2}

Therefore,

⇒ a^{2} = 5

⇒ b^{2} = 10 - a^{2}

⇒ b^{2} = 10 - 5

⇒ b^{2} = 5

Thus, the equation of the hyperbola is y^{2}/5 - x^{2}/5 = 1

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 y^{2}/5 - x^{2}/5 = 1 while the foci (0, ± √10) pass-through (2, 3)

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