ANSWER
D.
[tex]9 {y}^{2} - 4 {x}^{2}- 36y + 16x - 16 = 0[/tex]
EXPLANATION
The standard equation of the hyperbola is
[tex] \frac{ {(y - 2)}^{2} }{4} - \frac{ {(x - 2)}^{2} }{9} = 1[/tex]
We multiply through by 36 to obtain:
[tex]9 {(y - 2)}^{2} - 4( {x - 2)}^{2} = 36[/tex]
We now expand to get,
[tex]9( {y}^{2} - 4y + 4) - 4( {x}^{2} - 4x + 4) = 36[/tex]
Expand :
[tex]9 {y}^{2} - 36y + 36 - 4 {x}^{2} + 16x - 16 = 36[/tex]
To get the general form, we equate everything to zero to get,
[tex]9 {y}^{2} - 4 {x}^{2}- 36y + 16x - 16 = 0[/tex]
The correct choice is D.
