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

foci , passing through (2, 3)

Given: passing through (2, 3)

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 (0, ±be)

According to the question:

⇒ b²e² = 10

Since (2, 3) passing through hyperbola

Therefore,

{∵ a² = b²(e² – 1)}

⇒ 90 – 13b² = (10 – b²)b²

⇒ 90 – 13b² = 10b² – b⁴

⇒ 90 – 13b² – 10b² + b⁴ = 0

⇒ b⁴ – 23b² + 90 = 0

⇒ b⁴ – 18b² – 5b² + 90 = 0

⇒ b²(b² – 18) – 5(b² – 18) = 0

⇒ (b² – 18)(b² – 5) = 0

⇒ b² = 18 or 5

Case 1: b² = 18 and b²e² = 10

a² = b²(e² – 1)

⇒ a² = b²e² – b²

⇒ a² = 10 – 18

⇒ a² = – 8

Hence, equation of hyperbola is:

Case 2: b² = 5 and b²e² = 10

a² = b²(e² – 1)

⇒ a² = b²e² – b²

⇒ a² = 10 – 5

⇒ a² = 5

Hence, equation of hyperbola is: