Here foci are (0, ± √10) which lie on y-axis.
So the equation of hyperbola in standard form is

y2

a2

−

x2

b2

= 1
Now, foci are (0, ± √10) ⇒ c = √10
We know that c2 = a2+b2
⇒ (√10)2 = a2+b2 ⇒ b2 = 10 - a2
Since the hyperbola passes through (2, 3)
∴

9

a2

−

4

b2

= 1 ⇒

9

a2

−

4

10−a2

= 1
⇒ 9(10 − a2) − 4a2 − a2(10−a2) = 0
⇒ a4–23a2 + 90 = 0 ⇒ a4–18a2–5a2 + 90 = 0
⇒ (a2 – 18)(a2 – 5) = 0 ⇒ a2 = 18 or a2 = 5
When a2 = 18 then b2 = 10 – 18 = –8 (which is not possible)
When a2 = 5, then b2 = 10 – 5 = 5
Thus, required equation of hyperbola is

y2

5

−

x2

5

= 1.