How can I tell whether an ellipse is a circle from its general equation?

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Answer 1 · 2015-07-16T17:39:09

A circle in general form has the same non-zero coefficients for the $x^2$ and the $y^2$ terms. So if there is a graph, it is a circle (or a point).

Don't be too hasty, though.

$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$

Assuming that there is ineed a graph, it is:

an ellipse if $A$ and $C$ have the same sign.

a circle if $A=C$ .

However it is possible that there is no graph:

$x^2+y^2=-9/4$ Has no graph, but it can be rewritten as:

$4x^2+4y^2+9=0$ . At first look, this appears to be the equation of a circle, but it is not.