# How do you tell if an equation makes a circle?

For instance the equation, $$(x-3)^2+(y+4)^2=25$$ would be graphed as a circle. Is there a way to see that an equation creates a circle without graphing it? Is there a certain set of criteria it must fit or a generic base equation, or is it just random? Thank you.

• It definitely ain't random. If you know the distance formula, this just says the distance from a point to the center is a constant, which is exactly what makes a circle a circle. – Randall Mar 29 at 2:05
• I mean, $(x-h)^2 + (y-k)^2 = r^2$ is the "base equation" for a circle, centered at $(h,k)$, with radius $r$. I'm not exactly sure what you're trying to ask. – Eevee Trainer Mar 29 at 2:05

Take a general equation of a curve $$ax^2+2hxy+by^2+2gx+2fy+c=0$$ If $$abc+2fgh-af^2-bg^2-ch^2 \not=0$$ ; $$ab-h^2>0$$ ; $$a=b$$ and $$h=0$$; We can say that $$ax^2+2hxy+by^2+2gx+2fy+c=0$$ is a circle.
Or else as stated in comments,from the distance formula we deduce that,the general form of a circle of radius $$r$$ centered at $$(h,k)$$ is $$(x−h)^2+(y−k)^2=r^2$$