SAT Math: Equations of Circles

The focus of the new SAT has shifted more towards algebra and functions and away from geometry. However, you can still expect to see some questions on topics from geometry, including circles. You should know how to work with the equation of a circle in both its forms.

Standard Form: (x – h)² + (y – k)² = r²
General Form: x² + y² + Cx + Dy + E = 0

In the Standard Form, (hk) is the center of the circle and r is the radius.

The questions that give you the equation in Standard Form should be easier than those that involve General Form. Consider, for example, the following problem.

Circles 5-2A

(Note: Click the image to enlarge)

As long as you know that (hk) is the center of the circle and you’re careful with signs, you can quickly see that the center of this circle is at (3, -4). Additionally, if r² = 4, then the radius of the circle is 2. The correct answer is Choice B.

The questions become much harder if you are given the equation in General Form. Consider these two questions.

Circle 5-2B
(Note: Click the image to enlarge)

To answer these questions, you’ll need to remember how to complete the square to get the equation into standard form. This isn’t something you would have been expected to do on the “old” SAT, but this new version of the test has kicked it up a notch.

You should start by rearranging the problem so that the x terms and the y terms are together. You’ll also want to leave a couple of blanks that you’ll fill in with a number that “completes the square” for each of two trinomials.

x² – 4x + ___ + y² + 2y + ___ = 20

To find the number that goes in the first blank, cut the coefficient of x, which is 4, in half and then square the result. That’s 4. For the second blank, cut the coefficient of y, which is 2, in half and square what you get. That’s 1. Put those numbers in the blank, but be careful! This is an equation, so if we add something to one side, we need to do the same on the other side as well.

x² – 4x + 4 + y² + 2y + 1 = 20 + 4 + 1
x² – 4x + 4 + y² + 2y + 1 = 25

Now, let’s group together the two trinomials in the equation and factor.

(x²4x + 4) + (y² + 2y + 1) = 25
(x – 2)(x – 2) + (y + 1)(y + 1) = 25
(x – 2)² + (y + 1)² = 25

Now that the equation is in standard form, we can identify the center and radius. This circle has its center at (2, -1) and its radius is 5. If the radius is 5, then the diameter is 10. The correct answers to the two questions are
25)  C
26)  B

If you have any questions about these questions or anything to do with the SAT, send us an email at info@cardinalec.com

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