How can you tell if its an ellipse, circle or a parobola?

I noticed that the Parobla only has one term with a degree of 2 .

I noticed that the circle has same coefficients with their terms like y=x^2 +y^2 another example of cirlce y=9x^2 +9x^2

The Ellipse has different number coefficents that are not the same value like this. 4x^2 +5y^2

Does this sound about right? Im just seeing whats the difference so when the test question comes about identifying I will be equipped and not decieved.