Equations of circles

Connect centers and radii with equations and determine intersections and tangents.

Meaning

A circle equation expresses a fixed distance from a center. For center and radius it is

Example

has center and radius 5. is 5 units from the center and lies on the circle.From the center (−1,2) to (2,6), the coordinate differences are 3 and 4. Their squared sum 3²+4²=25 equals the squared radius, confirming the point lies on the circle.The right side is the squared radius, not the radius. A negative right side after completing squares gives no circle in the real plane.

Practice for this concept

Scope and references

Topics

Centers and radii, line–circle positions, tangents

Learning goal

Connect centers and radii with equations and determine intersections and tangents.

Helpful background

Check coordinate distance, circle centers, and radii.