6.4 Equations of Circles & Completing the Square

The Standard Equation of a Circle

Standard GEO-G.GPE.1 requires you to derive and use the equation of a circle and to complete the square to recover its center and radius. Every point on a circle is the same distance r from the center (h, k). Applying the distance formula, sqrt((x - h) squared + (y - k) squared) = r, and squaring both sides (which is exactly the Pythagorean theorem applied to the horizontal and vertical legs) gives the standard form:

(x - h) squared + (y - k) squared = r squared

Here (h, k) is the center and r is the radius. A circle centered at the origin simplifies to x squared + y squared = r squared. Understanding that the equation is just "distance from center equals radius" is what lets you both write an equation and read one.

Reading Center and Radius

Two sign and square-root details cause most Regents errors here.

• Signs flip. The equation subtracts the center coordinates, so a center of (3, -2) produces (x - 3) squared + (y + 2) squared = r squared. Read the opposite sign of what appears inside each parenthesis.
• Radius is a square root. The right side is r squared, not r. If (x - 3) squared + (y + 2) squared = 25, then r squared = 25 and the radius is 5. If x squared + y squared = 20, then r = sqrt(20) = 2 sqrt(5).

Writing an Equation From a Graph

Read the center off the graph, count the radius to any point directly right, left, up, or down, then substitute. A circle centered at (-2, 5) with radius 3 gives (x + 2) squared + (y - 5) squared = 9, because 3 squared = 9 and the center coordinates change sign inside the parentheses.

If the circle instead passes through a labeled point rather than showing the radius, use the distance formula between the center and that point to get r, or plug the point into the equation to solve for r squared. For example, a circle centered at (1, 2) that passes through (4, 6) has r = sqrt((4 - 1) squared + (6 - 2) squared) = sqrt(9 + 16) = 5, giving (x - 1) squared + (y - 2) squared = 25.

General Form and Completing the Square

An expanded circle appears in general form: x squared + y squared + Dx + Ey + F = 0. To find the center and radius, convert it back to standard form by completing the square on the x-terms and the y-terms separately.

The procedure:

1. Group the x-terms together and the y-terms together, and move the constant to the right side.
2. For each group, take half of the linear coefficient, square it, and add that number to BOTH sides.
3. Factor each group into a perfect square, then read the center and radius.

Worked example 1. The equation x squared + y squared - 6x + 4y = 12 is a circle. Half of -6 is -3, squared is 9; half of 4 is 2, squared is 4. Add 9 and 4 to both sides:

(x squared - 6x + 9) + (y squared + 4y + 4) = 12 + 9 + 4, which factors to (x - 3) squared + (y + 2) squared = 25. The center is (3, -2) and the radius is 5.

Worked example 2. For x squared + y squared + 8x - 10y + 16 = 0, first move the 16: x squared + 8x + y squared - 10y = -16. Half of 8 squared is 16; half of -10 squared is 25. Add both to each side:

(x + 4) squared + (y - 5) squared = -16 + 16 + 25 = 25, so the center is (-4, 5) and the radius is 5.

Checking That an Equation Is Really a Circle

After completing the square, the right side must be a positive number for a genuine circle. If the right side comes out to 0, the equation describes a single point (a degenerate circle of radius 0); if it is negative, no real graph exists. On the Regents you will almost always land on a positive r squared, but confirming the sign is a quick sanity check that your square-completing arithmetic is correct. Similarly, verify your center by substituting it back: the center coordinates should make each squared term equal 0.

Common Traps

• Forgetting to balance both sides. Whatever you add to complete a square must be added to the right side too, or the radius will be wrong.
• Sign of the center. (x + 4) squared means the x-coordinate of the center is -4, not +4.
• Radius vs. radius squared. The number on the right is r squared; take the square root to get the radius, and simplify radicals such as sqrt(20) = 2 sqrt(5).