Characteristic of Quadratic Equation that Represents Circle

Theorem
A quadratic equation in $2$ variables:


 * $a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$

describes a circle embedded in the Cartesian plane :


 * $(1): \quad h = 0$


 * $(2): \quad a = b \ne 0$

Necessary Condition
Consider the equation of a circle:


 * $A \paren {x^2 + y^2} + B x + C y + D = 0$

which is the equation of a circle with radius $R$ and center $\tuple {a, b}$, where:
 * $R = \dfrac 1 {2 A} \sqrt {B^2 + C^2 - 4 A D}$
 * $\tuple {a, b} = \tuple {\dfrac {-B} {2 A}, \dfrac {-C} {2 A} }$

provided:
 * $A > 0$
 * $B^2 + C^2 \ge 4 A D$

Thus it is seen that the coefficient for $x^2$ is the same as that for $y^2$, and there is no $x y$ term.

As can be seen, this matches the form as described: $a = b$ and $h = 0$.

Sufficient Condition
Consider the general quadratic equation in $2$ variables:


 * $a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$

Let us set $a = b$ and $h = 0$.

We obtain:


 * $a x^2 + a y^2 + 2 g x + 2 f y + c = 0$

Assuming $a \ne 0$, we get:

It is seen that $(1)$ is Equation of Circle in Cartesian Plane, where:


 * the represents the distance from $\tuple {x, y}$ to the point $\tuple {-\dfrac g a, -\dfrac f a}$ which is its center


 * the represents the radius $\sqrt {\dfrac {g^2 + f^2 - a c} {a^2} }$.

If $a$, $g$, $f$ and $c$ are all real numbers, then the center is always a point in the cartesian plane.

The radius, however, is real only if $g^2 + f^2 > a c$.

If $g^2 + f^2 < a c$, then no real $x$ and $y$ can satisfy $(1)$, as $\paren {x + \dfrac g a}^2 + \paren {y + \dfrac f a}^2 + \dfrac {a c - g^2 + f^2} {a^2}$ is always strictly positive.

Hence the locus of $(1)$ is a virtual circle.

If $g^2 + f^2 = a c$, then $(1)$ degenerates to:
 * $\paren {x + \dfrac g a}^2 + \paren {y + \dfrac f a}^2 = 0$

which is satisfied for $\tuple {x, y} = \tuple {-\dfrac g a, -\dfrac f a}$ only.

Hence the radius is $0$ and the locus of $(1)$ is a point-circle.

If $a = 0$ then the general quadratic equation degenerates to:


 * $2 g x + 2 f y + c = 0$

which is the equation of a straight line.