Circles

A circle in the coordinate plane is defined as the set of all points that lie at a constant distance, known as the radius, from a fixed point called the center. This relationship is captured using the distance formula. For a point (x, y) on the circle and a center (h, k), the distance between them equals the radius r. By squaring both sides of the distance formula, the equation of the circle is written in standard form:

Constructing the Equation from Geometric Information

If the center and the radius of a circle are known, the equation can be directly formed using substitution. Consider a circle with its center at (1, 2) and radius 4. Substituting into the standard form yields:

This equation represents all points (x, y) that lie 4 units from the point (1, 2). This simple approach is often sufficient for constructing equations when the geometric characteristics of the circle are explicitly given.

Identifying the Circle from a General Equation

When the circle’s equation is not presented in standard form, completing the square helps in identifying its center and radius. Take the equation

Group and complete the square:

This reveals the center at (3, 2) and a radius of 4. Rewriting equations in standard form is essential for graphing and analyzing circles effectively.