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

Question

5172 views around the world

You can reuse this answer
Creative Commons License

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.