Circles are a classic SAT topic—sometimes they show up as equations, sometimes as tangents or geometry diagrams. You’ll need to complete the square, find centers and radii, and sometimes connect slopes for tangents. This walkthrough covers everything you need for SAT circle mastery!
equation of a circle
The general equation for a circle with center (h,k) and radius r is:
(x−h)2+(y−k)2=r2
To find the center and radius, rewrite the equation in this form—usually by completing the square.
completing the square (with example)
Completing the square lets you rewrite circle equations and easily find the center and radius.
For example:
Given: x2+8x+y2−6y=0
x2+8x+y2−6y=0
Group and complete the square for each variable:
(x2+8x)+(y2−6y)=0
(x2+8x+16)+(y2−6y+9)=0+16+9
(x+4)2+(y−3)2=25
The center is (–4, 3), the radius is 5.
tangents and slopes
The tangent to a circle at a point is perpendicular to the radius at that point. If you know the radius’s slope, the tangent’s slope is the negative reciprocal.
For example:
If the center is (−4,−6) and the tangent touches at (−7,−7), find the slope of the radius:
slope=−7−(−4)−7−(−6)=−3−1=31
The tangent's slope is −3 (negative reciprocal).
circles with coefficients
If the circle equation has coefficients in front of x2 or y2, factor them out first:
For example:
2x2−6x+2y2+2y=45
x2−3x+y2+y=22.5
Now, complete the square for each variable.
Practice with these real SAT-style circle questions!
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Question 1
x2+20x+y2+16y=−20
The equation above defines a circle in the xy-plane. What are the coordinates of the center of the circle?
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Question 2
A circle in the xy-plane has its center at (−4,−6). Line k is tangent to this circle at the point (−7,−7). What is the slope of line k?