A circle with center (h,k) and radius r is (x−h)2+(y−k)2=r2. From x2+y2+Dx+Ey=F, complete the square: add (2D)2 and (2E)2 to both sides. Divide out any common coefficient on x2 and y2 first.
The trap
Forgetting to add the completing-the-square terms to the right side, or to divide by the coefficient of x2 first.
Can Desmos do it?
Desmos draws the circle, and you can read the center if it lands on grid lines. With a missing coefficient, completing the square is faster and exact.
College Board skill: Circles · Geometry and Trigonometry
Try one at each level
One question from each level. The drill has 12 at every level, and moves you up after four right in a row.
Level 1 · Warm-up
In the xy-plane, the graph of x2+y2+12x−8y=−3 is a circle. What is the radius of the circle?
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Answer: 7Complete the square: (x+6)2+(y−4)2=−3+36+16=49. The radius is 49=7.
Level 2 · Test level
In the xy-plane, the graph of 4x2+4y2+24x+8y=60 is a circle. What is the radius of the circle?
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Answer: 5Divide every term by 4 first: x2+y2+6x+2y=15. Completing the square: (x+3)2+(y+1)2=15+9+1=25, so the radius is 5. Skipping the division would give 70, which is wrong.
Level 3 · Beyond the test
In the xy-plane, the graph of x2+y2−8x+by=−48, where b is a constant, is a circle with radius 2. If b<0, what is the y-coordinate of the center of the circle?
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Answer: 6Complete the square: (x−4)2+(y+2b)2=−48+16+4b2. Setting this equal to 4: 4b2=36, so b=±12; the sign condition gives b=−12. The center's y-coordinate is −2b=6.
Practice all 36 questions
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