When A=CA = CA=C, the equation Ax2+Cy2+Dx+Ey+F=0A x^{2} + C y^{2} + D x + E y + F = 0Ax2+Cy2+Dx+Ey+F=0 simplifies to A(x2+y2)+Dx+Ey+F=0A(x^{2} + y^{2}) + D x + E y + F = 0A(x2+y2)+Dx+Ey+F=0. This represents a circle because the coefficients of x2x^{2}x2 and y2y^{2}y2 are equal and nonzero.
When AC=0A C = 0AC=0, either A=0A = 0A=0 or C=0C = 0C=0. If A=0A = 0A=0, the equation becomes Cy2+Dx+Ey+F=0C y^{2} + D x + E y + F = 0Cy2+Dx+Ey+F=0, which is a parabola. Similarly, if C=0C = 0C=0, the equation becomes Ax2+Dx+Ey+F=0A x^{2} + D x + E y + F = 0Ax2+Dx+Ey+F=0, which is also a parabola. Thus, when AC=0A C = 0AC=0, the conic is a parabola.
When A≠CA \neq CA=C and AC>0A C > 0AC>0, the coefficients of x2x^{2}x2 and y2y^{2}y2 are both nonzero and have the same sign. This indicates that the conic is an ellipse. If A=CA = CA=C, it would be a circle, but since A≠CA \neq CA=C, it is an ellipse.
If A=CA = CA=C, then the answer is a circle\boxed{\text{a circle}}a circle.
If AC=0A C = 0AC=0, then the answer is a parabola\boxed{\text{a parabola}}a parabola.
If A≠CA \neq CA=C and AC>0A C > 0AC>0, then the answer is an ellipse\boxed{\text{an ellipse}}an ellipse.
If AC<0A C < 0AC<0, then the answer is a hyperbola\boxed{\text{a hyperbola}}a hyperbola.
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