Ax²+Bx+Cy²+Dy+E=0
so in the template above for a conic, the defining components are the coefficients A and C
If either A=0 or C=0, then the equation is a parabola
if A=C, then we have a Circle
if either A or C is negative, and A≠C, then we have a hyperbola
if A and C are both positive or both negative, and A≠C, then we have an ellipse.