When an equation is a circle

Not every second-degree equation in $x$ and $y$ represents a circle. One must check the structure of the degree-two terms and the sign of the squared radius.

Caution — When an equation is not a circle

For an equation of the type $Ax^2 + By^2 + Cx + Dy + E = 0$ to represent a circle it is necessary that:

• the terms $x^2$ and $y^2$ are present simultaneously (otherwise it is a parabola or it degenerates);
• the coefficients of $x^2$ and $y^2$ are equal ($A = B$, so that dividing by $A$ gives the standard form);
• $R^2 = \tfrac{a^2}{4} + \tfrac{b^2}{4} - c$ is positive. If $R^2 = 0$ the “equation” describes a single point (the centre); if $R^2 < 0$ it has no real solutions (empty set).

The cases $R^2 = 0$ and $R^2 < 0$ are called degenerate circles: formally the equation is in standard form, but geometrically there is no genuine circle.

Links

Topics: Analytical circle
Concepts: Degenerate circle · Circle equation · Standard form · Radius
Skills: Analytical geometry · Reasoning by cases