Question

is equation (y-4)^2=9(x-4) is it a circle, ellipse, parabola or hyperbola?

Answer 1

It's a parabola, but it sitting on it side so to speak.

\[(y-4)^2=9(x-4) -> x = 52/9 - (8 y)/9 + y^2/9\]

you see you got a y^2 term so its a parabola, but its in the direction of the x axis:
http://www.wolframalpha.com/input/?i=(y-4)^2%3D9(x-4)

Answer 2

Circles, ellipses, and hyperbolas all have an x square term AND a y square term. This equation only has a y square term, so it is a parabola.