11.4 Circles in the Coordinate Plane

11.4 Circles in the Coordinate Plane

College Board’s Next-Generation ACCUPLACER Advanced Algebra and Functions (AAF) test weights Geometry concepts for Algebra 2 at 5–10%, typically 1–2 CAT items. Table 11’s last Algebra 2 geometry skill is circle equations in the coordinate plane. The standard form is

(x − h)^2 + (y − k)^2 = r^2

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Reading standard form

(x − 3)^2 + (y + 2)^2 = 16

Center (3, −2), because y + 2 = y − (−2), and r^2 = 16 so r = 4. Radius is never negative. r^2 = 16 does not make r = −4.

Trap: flipping the sign of h or k. (x + 1)^2 is center x-coordinate −1, not +1. Trap: reporting r = 16 or r = 9 (forgetting to square-root). Trap: writing the right-hand side as the radius when the stem asks for r.

Worked: write the equation from a center and a point

Center (2, −1), circle passes through (5, 3).

Radius is the distance: r^2 = (5 − 2)^2 + (3 − (−1))^2 = 9 + 16 = 25.

Equation: (x − 2)^2 + (y + 1)^2 = 25.

You can leave r as 5, but the equation wants r^2 on the right side. Do not write = 5 on the right; that would be r, not r^2.

Worked: diameter endpoints

A circle has diameter from A(−1, 4) to B(5, −2). The center is the midpoint:

h = (−1 + 5)/2 = 2, k = (4 + (−2))/2 = 1, so center (2, 1).

r^2 is the squared distance from the center to A (or half the diameter, squared):

(−1 − 2)^2 + (4 − 1)^2 = 9 + 9 = 18

Equation: (x − 2)^2 + (y − 1)^2 = 18. Radius is √18 = 3√2, but leaving r^2 = 18 in the equation is the standard-form requirement.

Completing the square: the AAF workhorse

General form: x^2 + y^2 + Dx + Ey + F = 0

Group x terms, group y terms, move the constant, complete both squares.

Worked: x^2 + y^2 − 6x + 4y − 3 = 0

Group: (x^2 − 6x) + (y^2 + 4y) = 3

Complete x: take half of −6, which is −3, square to 9. Add 9.

Complete y: take half of +4, which is +2, square to 4. Add 4.

Add 9 and 4 to both sides:

(x^2 − 6x + 9) + (y^2 + 4y + 4) = 3 + 9 + 4

(x − 3)^2 + (y + 2)^2 = 16

Center (3, −2), radius 4.

The linear x-coefficient is −6, so h = −D/2 = 6/2 = 3 if you write general form as x^2 + y^2 + Dx + Ey + F = 0 with D = −6. The linear y-coefficient is +4, so k = −E/2 = −2. Same result as completing the square; use whichever you can do without dropping a sign.

Trap: forgetting to add 9 and 4 to the right side, leaving = 3 and claiming r = √3. Trap: (y − 2)^2 from the +4y term — half of +4 is +2, so (y + 2)^2, center y-coordinate −2. Trap: radius 16. Trap: moving −3 to the right as −3 instead of +3.

Walk the constant one more time, because that is where most CAT misses happen. Starting from x^2 + y^2 − 6x + 4y − 3 = 0, add 3 to both sides first: x^2 − 6x + y^2 + 4y = 3. Then add 9 and 4. The right side is 3 + 9 + 4 = 16, not −3 + 9 + 4. The original −3 is a left-side constant that becomes +3 on the right before the completed-square constants join it.

Another worked: x^2 + y^2 + 8x − 2y + 13 = 0

(x^2 + 8x) + (y^2 − 2y) = −13

Half of 8 is 4, square 16. Half of −2 is −1, square 1.

(x + 4)^2 + (y − 1)^2 = −13 + 16 + 1 = 4

Center (−4, 1), radius 2.

If the right side came out negative, there is no real circle (empty set). If it came out zero, the graph is a single point (degenerate circle of radius 0). AAF multiple-choice usually gives a genuine circle.

Worked: a missing y-linear term

x^2 + y^2 − 10x + 21 = 0

(x^2 − 10x) + y^2 = −21

Complete only x: half of −10 is −5, square 25.

(x − 5)^2 + y^2 = −21 + 25 = 4

Center (5, 0), radius 2. When a linear term is missing, that coordinate of the center is 0. Do not invent a (y − 21) shift from the constant term.

Inside, on, or outside

Compare the distance from the center to a point with the radius.

• Distance < r: inside
• Distance = r: on the circle
• Distance > r: outside

You can compare squared distances to r^2 to avoid a square root. That is the AAF-friendly test, because the standard form already hands you r^2.

Worked: the circle (x − 3)^2 + (y + 2)^2 = 16, several points

Center C = (3, −2), r = 4, r^2 = 16. This is the completed-square form of x^2 + y^2 − 6x + 4y − 3 = 0.

Point A (6, 2): (6 − 3)^2 + (2 − (−2))^2 = 9 + 16 = 25. 25 > 16, so A is outside. Distance is 5, which is greater than 4.

Point B (3, 2): (3 − 3)^2 + (2 + 2)^2 = 0 + 16 = 16. 16 = 16, so B is on the circle. Distance 4.

Point C (4, −2): (4 − 3)^2 + (−2 + 2)^2 = 1 + 0 = 1. 1 < 16, so C is inside. Distance 1.

Point D (3, −2): distance 0, the center itself, inside (strictly inside for r > 0).

A stem may ask which listed point lies on the circle, or which is farthest from the center. Compute d^2 for each; the largest d^2 is farthest. Here A is farthest of the four, then B on the circle, then C, then the center.

Worked: a point given before the equation is in standard form

Is (1, 1) inside the circle x^2 + y^2 − 6x + 4y − 3 = 0?

You already completed the square: center (3, −2), r^2 = 16.

(1 − 3)^2 + (1 + 2)^2 = 4 + 9 = 13. 13 < 16, so inside.

You can also plug (1, 1) into the general form: 1 + 1 − 6 + 4 − 3 = −3. Interpreting the sign of the general form is messier than completing the square first; on AAF, convert to standard form, then compare d^2 to r^2.

Expanding standard form (the reverse)

Sometimes options are in general form and the stem gives center and radius.

Center (3, −2), radius 4:

(x − 3)^2 + (y + 2)^2 = 16

x^2 − 6x + 9 + y^2 + 4y + 4 = 16

x^2 + y^2 − 6x + 4y + 13 = 16

x^2 + y^2 − 6x + 4y − 3 = 0

That is the original equation. Expanding is how you match an option written as x^2 + y^2 + Dx + Ey + F = 0. Completing the square is how you go the other way. If two options differ only in the constant, they came from adding the completed-square numbers to the wrong side.

AAF circle checklist

1. Standard form (x − h)^2 + (y − k)^2 = r^2: center (h, k), radius √(right-hand side).
2. (x + a)^2 means h = −a. (y + b)^2 means k = −b.
3. Complete the square on x and on y. Add the same constants to the right side.
4. Radius is the square root of r^2, never r^2 itself, and never negative.
5. Classify a point by comparing (x − h)^2 + (y − k)^2 to r^2.
6. Diameter endpoints: center is the midpoint; r^2 is the squared distance from the center to an endpoint.

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