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