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.
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 $(x−h)^2+(y−k)^2=r^2$