Equations of Circles
Overview
The equation of a circle is the Pythagorean Theorem in disguise: every point (x, y) on the circle is exactly r units from the center (h, k), so (x − h)² + (y − k)² = r². Reading an equation is therefore mechanical — the center is (h, k) with the signs switched from the way they appear, and the radius is the square root of the number on the right. Writing an equation runs the machine in reverse, using the distance formula when a point on the circle is given instead of the radius.
Vocabulary
- Standard Form of the Equation of a Circle
- (x − h)² + (y − k)² = r², where (h, k) is the center of the circle and r is its radius.
- Center
- The point (h, k) in the standard form. The signs in the equation are switched: (x + 1)² means h = −1.
- Radius
- The distance from the center to any point on the circle. In standard form, r² appears on the right side, so r is the square root of that value.
- General Form
- The expanded form x² + y² + 2gx + 2fy + c = 0. Completing the square on x and y converts it to standard form, revealing the center (−g, −f) and the radius.
Main Ideas
- For a circle centered at the origin, the equation is simply x² + y² = r².
- Read the center from (x − h)² + (y − k)² by switching both signs: (x − 3)² + (y + 2)² has center (3, −2).
- The number on the right is r², not r. If the right side is 25, the radius is 5.
- Given a center and a point on the circle, find r with the distance formula first, then substitute into standard form.
- Given a general-form equation, complete the square on both variables to reach standard form before reading off the center and radius.
Lesson Notes
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Practice Problems
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