Equations of Circles and Completing the Square

Objective: Read a center and radius from a circle equation.

Prerequisite skills: distance formula and completing the square. A circle consists of points a fixed distance r from center (h,k). Squaring the distance equation yields (x−h)²+(y−k)²=r². The displayed signs inside the squares are opposite the center coordinates. Common error: reading r² as r or reversing a center sign. Restrictions and interpretation: In standard circle form the right side is radius squared; a positive value gives a real nondegenerate circle.

Objective: Convert a general circle equation to standard form.

Group x-terms and y-terms separately. Add the square of half each linear coefficient on both sides, then rewrite both squares. The new right side is the radius squared. Common error: adding completing constants on the left but not the right. Restrictions and interpretation: In standard circle form the right side is radius squared; a positive value gives a real nondegenerate circle.

Objective: Use a given center-point distance to determine a circle constant.

If a point lies on a circle, its squared distance to the center equals r². Substitute the coordinates with grouped differences. This can determine a radius or a missing equation constant exactly. Common error: forgetting to subtract the center coordinates before squaring. Restrictions and interpretation: In standard circle form the right side is radius squared; a positive value gives a real nondegenerate circle. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.