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

Jul 16, 2015

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).

#### Explanation:

Don't be too hasty, though.

$A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$

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:

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

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