Find the equations of the hyperbola satisfying the given conditions :Foci `(0,+-sqrt(10))`, passing through (2,3)

Explanation

Let equation of hyperbola is
`(y^(2))/(b^(2))-(x^(2))/(a^(2))=1` . . . (1) `(because` foci lie on y-axis)
Coordinates of foci `=(0,pmsqrt(10))`
`rArr" "be=sqrt(10)`
`rArr" "b^(2)e^(2)=10`
`rArra^(2)+b^(2)=10` . . .(2)
Hyperbola passes through the point (2,3).
Therefore from equation (1)
`(9)/(b^(2))-(4)/(a^(2))=1`
`(9)/(b^(2))-(4)/(10-b^(2))=1` [From equation (2)]
`rArr(90-9b^(2)-ab^(2))/(b^(2)(10-b^(2)))`
`rArr""90-13b^(2)=10b^(2)-b^(4)`
`rArr""b^(4)-23b^(2)+90=0`
`(b^(2)-5)(b^(2)-18)=0`
`rArr""b^(2)=5orb^(2)=18`
From equation (2)
`b^(2)=5rArra^(2)=5`
`b^(2)=18rArra^(2)=-8` which is not possible
`:." "b^(2)=a^(2)=5`
and equation of hyperbola
`(y^(2))/(5)-(x^(2))/(5)=1rArry^(2)-x^(2)=5`