SAT Math: Circle Equations and Coordinates Practice Questions
Read a circle's center and radius, complete the square, and use distances to check points and tangents. Try each original question before checking its worked explanation. This focused practice is free and requires no account.
Free original practice questions for study purposes. Open Exam Prep is an independent study resource and is not affiliated with, endorsed by, or sponsored by the makers of SAT.
Answer the questions below — you get instant feedback and a full explanation for each.
1. What is the center of the circle (x - 3)^2 + (y + 4)^2 = 49?
Explanation. In (x - h)^2 + (y - k)^2 = r^2, the center is (h, k). Here y + 4 equals y - (-4), so the center is (3, -4).
2. The circle (x + 2)^2 + (y - 5)^2 = 81 has what radius?
Explanation. The right side equals the square of the radius, not the radius itself. Since r^2 = 81 and a radius is positive, r = 9.
3. What is the radius of the circle x^2 + y^2 - 8x + 6y = 11?
Explanation. Complete the square: (x - 4)^2 - 16 + (y + 3)^2 - 9 = 11. Therefore (x - 4)^2 + (y + 3)^2 = 36, giving radius 6.
4. A circle has center (2, -1) and is tangent to the vertical line x = 7. What is its radius?
Explanation. The perpendicular distance from (2, -1) to x = 7 is |7 - 2| = 5. A radius to the point of tangency is perpendicular to the tangent, so this distance is the radius.
5. Which point lies on the circle (x - 2)^2 + (y + 1)^2 = 25?
Explanation. For (7, -1), the left side is 5^2 + 0^2 = 25. The other points give 16, 20, and 26 respectively, so only (7, -1) lies on the circle.
6. The endpoints of a diameter are (-1, 2) and (5, 10). What is the radius of the circle?
Explanation. The diameter length is sqrt((5 + 1)^2 + (10 - 2)^2) = sqrt(36 + 64) = 10. The radius is half the diameter, so it is 5.
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FAQ
Why do the signs in a circle's center look reversed?
The standard form uses x minus the center's x-coordinate and y minus the center's y-coordinate. For example, y + 4 means the center has y-coordinate -4.
When do I complete the square?
Complete the square when an equation contains separate x-squared and y-squared terms with linear terms. The resulting standard form reveals the center and radius.