Step 1: Assuming equation of hyperbola
Given:
Foci are (0,±√10)and the hyperbola is passing through (2,3)
As we can see that foci are on the y − axis.
So, the equation of the hyperbola will be of the form y2a2−x2b2=1 ...(1)
Step 2: Finding the values of a2 and b2
Foci =(0,±√10)=(0,±c) i.e., c=√10
As we know that c2=a2+b2
So, (√10)2=a2+b2
⇒a2+b2=10
⇒b2=10−a2 ...(2)
Equation (1) passing through (2,3)
⇒y2a2−x2b2=1
⇒32a2−22b2=1
⇒9a2−4b2=1 ....(3)
From (2) and (3)
⇒9a2−410−a2=1
⇒9(10−a2)−4a2a2(10−a2)=1
⇒90−9a2−4a2a2(10−a2)=1
⇒90−13a2=a2(10−a2)
⇒90−13a2=10a2−a4
⇒a4−23a2+90=0
Let a2=x
So, our equation becomes
⇒x2−23x+90=0
⇒x2−18x−5x+90=0
⇒x(x−18)−5(x−18)=0
⇒(x−18)(x−5)=0
⇒x=5,18
⇒a2=5,18
Putting these values in equation (2), we get
Case -1 : a2=5
⇒b2=10−5=5
Case-2 : a2=18
⇒b2=10−18=−8 (Not possible)
∴a2=5 and b2=5
Hence the required equation of the hyperbola is y25−x25=1