What curve is x^2 + (y-2)^2 = 4?

Question

I not only want to know what this curve is (parabola, circle, eclipse, or hyperbola), but I also want to know how you can tell.

Answer 1

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Answer 2

circle

If you expand and FOIL the (y-2)^2, you'll get x^2 + y^2 - 4y + 4 = 4. All you do is look at the signs and coefficients of the x^2 and y^2 terms. If:

they're equal and same signs -- circle

they're not equal and same signs -- ellipse

they're opposite signs -- hyperbola

A parabola will happen when there's only one squared term, like all the squared terms will be the same variable. e.g. if there's an x^2, there can't be any y^2 's