See post #2 here
See Conic section - Wikipedia, the free encyclopedia
When looking at it your equations disregard (for purposes of identification) the offsets on x and y. For example, whatever (x-3)^2/9 + (y-2)^2/4 =1 is, is just a shifted version of x^2/9 + y^2/4 =1 (shifted 3 to the right and 2 up).