8.5 Circles: Center, Radius & the Equation of a Circle
Key Takeaways
- The standard equation of a circle with center (h, k) and radius r is (x − h)² + (y − k)² = r², a direct application of the distance formula.
- The signs in the equation are opposite the coordinates of the center, so (x − 3)² + (y + 5)² = 16 has center (3, −5) and radius 4.
- The right side of the standard form is r², not r, so a right side of 49 corresponds to a radius of 7.
- General form x² + y² + Dx + Ey + F = 0 is converted to standard form by completing the square separately in x and in y.
- A point lies inside, on, or outside a circle according to whether its distance from the center is less than, equal to, or greater than the radius.
8.5 Circles: Center, Radius & the Equation of a Circle
Skill 11 of Competency 3 is a specific, self-contained topic: determine the center, the radius, and the equation of a circle, and select graphical representations of a circle on a coordinate plane. It rewards a small amount of memorization heavily.
The definition and the standard equation
A circle is the set of all points in a plane at a fixed distance — the radius — from a fixed point, the center.
Translate that definition through the distance formula. A point (x, y) lies on the circle centered at (h, k) with radius r exactly when
√((x − h)² + (y − k)²) = r
Squaring both sides gives the standard form:
(x − h)² + (y − k)² = r²
The equation is not a new fact to memorize; it is the distance formula with the radical cleared. Recognizing that makes the sign conventions inevitable rather than arbitrary.
Reading center and radius
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| (x - h)^2 + (y - k)^2 = r^2 center (h, k), radius r |
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| (x - 3)^2 + (y - 5)^2 = 49 -> center (3, 5), radius 7 |
| (x + 2)^2 + (y - 6)^2 = 25 -> center (-2, 6), radius 5 |
| (x + 4)^2 + (y + 1)^2 = 12 -> center (-4, -1), radius 2*sqrt(3) |
| x^2 + (y - 8)^2 = 1 -> center (0, 8), radius 1 |
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Two rules generate nearly every error on this skill:
- The center's coordinates are the opposite of the signs shown. The form subtracts h, so (x + 2)² is really (x − (−2))², and h = −2.
- The right side is r², so take a square root. A right side of 49 means r = 7, not 49. And a right side of 12 means r = √12 = 2√3, an irrational radius — perfectly legitimate.
Writing the equation from given information
From center and radius.
Center (−5, 2), radius 6: (x + 5)² + (y − 2)² = 36
From center and a point on the circle. Find the radius with the distance formula.
Center (1, −3) passing through (5, 0): r = √(16 + 9) = 5. Equation: (x − 1)² + (y + 3)² = 25
From the endpoints of a diameter. The center is the midpoint; the radius is half the diameter.
Diameter from (2, 3) to (10, 9). Center = midpoint = (6, 6). Diameter length = √(64 + 36) = 10, so r = 5. Equation: (x − 6)² + (y − 6)² = 25
Computing the radius as the full diameter is the standard slip here, producing a right side of 100.
Converting from general form
The general form is x² + y² + Dx + Ey + F = 0. Convert by completing the square in x and in y separately — the same technique used for quadratics in section 6.2.
Convert x² + y² − 6x + 10y + 18 = 0. Group and move the constant: (x² − 6x) + (y² + 10y) = −18 Complete both squares. Half of −6 is −3, and (−3)² = 9. Half of 10 is 5, and 5² = 25. Add both to each side: (x² − 6x + 9) + (y² + 10y + 25) = −18 + 9 + 25 (x − 3)² + (y + 5)² = 16 Center (3, −5), radius 4.
Note that both added constants must appear on the right side. Adding 9 and 25 on the left while adding only one of them on the right is a common failure.
A caution: if the right side comes out zero, the "circle" is the single center point. If it comes out negative, no real points satisfy the equation and the graph is empty.
Graphing and selecting a representation
To graph, plot the center and then move r units right, left, up, and down to get four points on the circle, then sketch through them.
(x + 1)² + (y − 4)² = 9: center (−1, 4), radius 3. Points at (2, 4), (−4, 4), (−1, 7), (−1, 1).
Because the skill says select graphical representations, items often show four graphs and ask which matches an equation. Check two things in order: where is the center, and how far does the curve reach. Distractors typically shift the center by reversing a sign or use a radius equal to the right side rather than its square root.
Determining whether a point is on, inside, or outside
Compare the point's distance from the center to the radius.
Circle (x − 2)² + (y + 1)² = 25, center (2, −1), radius 5. Where is the point (6, 2)? Distance = √((6 − 2)² + (2 − (−1))²) = √(16 + 9) = 5. Distance equals the radius, so the point lies on the circle.
Where is (0, 0)? Distance = √(4 + 1) = √5 ≈ 2.24 < 5, so the origin lies inside.
The algebraic shortcut is to substitute the point into the left side and compare with the right side: less than r² means inside, equal means on, greater means outside. Substituting (0, 0) gives 4 + 1 = 5 < 25, confirming the point is inside without extracting any square root.
What are the center and radius of the circle (x + 7)² + (y − 2)² = 45?
A circle has a diameter with endpoints (−2, 5) and (8, −1). What is its equation in standard form?
Convert x² + y² + 8x − 4y − 5 = 0 to standard form and identify the radius.