Respuesta :
We have been given an equation of hyperbola [tex]\frac{(y+3)^2}{81}-\frac{(x-6)^2}{89}=1[/tex]. We are asked to find the center of hyperbola.
We know that standard equation of a vertical hyperbola is in form [tex]\frac{(y-k)^2}{b^2}-\frac{(x-h)^2}{a^2}=1[/tex], where point (h,k) represents center of hyperbola.
Upon comparing our given equation with standard vertical hyperbola, we can see that the value of h is 6.
To find the value of k, we need to rewrite our equation as:
[tex]\frac{(y-(-3))^2}{81}-\frac{(x-6)^2}{89}=1[/tex]
Now we can see that value of k is [tex]-3[/tex]. Therefore, the vertex of given hyperbola will be at point [tex](6,-3)[/tex] and option D is the correct choice.
