SAT circle equations: center, radius, and tangency
Expanded circle equations test algebra and geometry at once: the center depends on signs, while the radius depends on keeping both sides of the equation balanced.
How do you find a circle’s center and radius from an expanded equation on the SAT?
Complete the square in x and y to rewrite the equation as (x − h)² + (y − k)² = r², then read the center as (h, k) and take the positive square root of r² for the radius.
Worked example: do not lose the added constants
Problem. Find the center and radius of x² + y² − 10x + 6y − 2 = 0.
- Group variables and move the constant: (x² − 10x) + (y² + 6y) = 2.
- Add 25 and 9 to both sides: (x² − 10x + 25) + (y² + 6y + 9) = 36.
- Factor: (x − 5)² + (y + 3)² = 36.
- The center is (5, −3) and the radius is 6.
Reporting 36 confuses radius squared with radius. Reporting (−5, 3) reverses the signs. Reading the center directly from −10x + 6y without halving the coefficients produces a third common error.
Turn the same equation into a tangent question
A horizontal tangent touches directly above or below the center, so its equation is y = k + r or y = k − r. In this example, the horizontal tangents are y = 3 and y = −9. The vertical tangents are x = 11 and x = −1. These four lines sit exactly one radius from the center.
Check that the equation represents a real circle
Normalize equal, nonzero x² and y² coefficients before completing the square. A positive value of r² gives a circle; r² = 0 gives a single point; a negative value gives no real points. Unequal coefficients on x² and y² generally describe a different conic, not a circle. Do not apply the radius formula until the equation has the right form.
When a graph is useful
Graph the original equation to check the location and approximate radius, then compare it with the completed-square form. The two equations should draw the same circle. If the question asks for an exact radius such as √13, keep the radical instead of rounding early.
Try it before you read the answer
1. Find the center and radius: x² + y² + 4x − 12y + 15 = 0.
Show explanation
Completing the square gives (x + 2)² + (y − 6)² = 25; center (−2, 6), radius 5.
2. What are the horizontal tangents to (x + 2)² + (y − 6)² = 25?
Show explanation
y = 11 and y = 1; add and subtract the radius 5 from the center’s y-coordinate 6.
Practice the decision, not just the arithmetic.
Select SAT Math and Geometry in VECTOR; after a circle problem, write the center and radius separately before calculating the requested distance, area, or tangent.
VECTOR offers a game-based alternative to a conventional practice session. Khan Academy provides official SAT preparation; use the tool that fits the skill you need to learn. No head-to-head score advantage is established.
Sources and scope
The mathematical examples above are independently written, not copied official exam questions. Exam scope and official-practice claims are checked against these sources: