Definition:Point-Circle

Definition

A point-circle is the locus in the Cartesian plane of an equation of the form:

$(1): \quad \paren {x - a}^2 + \paren {y - b}^2 = 0$

where $a$ and $b$ are real constants.

There is only one point in the Cartesian plane which satisfies $(1)$, and that is the point $\tuple {a, b}$.

It can be considered to be a circle whose radius is equal to zero.

Also known as

A point-circle is also known as a degenerate circle.

Also see

• Definition:Virtual Circle

• Characteristic of Quadratic Equation that Represents Circle‎

• Results about point-circles can be found here.

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $14$. To find the equation of the circle whose centre is $\tuple {\alpha, \beta}$ and radius $r$