Digital SAT · Math · Circles
Circle equations on the SAT
(x - h)^{2} + (y - k)^{2} = r^{2} is a circle with center (h, k) and radius r. Many SAT circle questions give the equation multiplied out instead; complete the square to read the center and radius.
Completing the square
Group the x terms and the y terms. For each group, take half the coefficient of the middle term, square it, and add it to both sides.
x^{2} + 6x + y^{2} - 4y = 12 becomes (x^{2} + 6x + 9) + (y^{2} - 4y + 4) = 12 + 9 + 4, or (x + 3)^{2} + (y - 2)^{2} = 25. The center is (−3, 2) and the radius is 5.
The traps
- Answering r² for r. The right side is the radius squared. 25 means a radius of 5.
- Reading the radius before completing the square. In the example, 12 is not r²; the 9 and 4 added to both sides belong to it.
- Flipping the signs of the center. (x + 3)^{2} means h is −3, not 3.
Practice, with new numbers every time
Circles
The equation x^{2} - 14x + y^{2} - 6y = 23 represents a circle in the xy-plane.
What is the length of the radius of the circle?