In each of the, find the equations of the hyperbola satisfying the given conditions.

Foci (0, ), passing through (2, 3)

Foci (0, ), passing through (2, 3)

Here, the foci are on y-axis.

Thus,

The equation of the hyperbola is of the form

Since, the foci are (, 0), c =

We know that, a² + b² = c²

⇒ b² = 10 – a² …………..(1)

Since, the hyperbola passes through point (2, 3)

……………(2)

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

⇒ 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 that is c²> a²

Thus, a² = 5

⇒ b² = 10 – a² = 10 – 5 = 5

Hence, the equation of the hyperbola is