Equations of Circles

Unit 12: Circles · ~15–30 min

Standard form

Back in Lesson 12.1, a circle was defined as every point the same distance from a center. On the coordinate plane, that distance statement translates directly into an equation using the distance formula. For a circle with center (h, k) and radius r, the standard form equation is:

Every point (x, y) that makes this equation true is exactly r units from (h, k) — which is precisely the definition of the circle.

Completing the square

Sometimes a circle's equation is given in expanded form, like x² + y² + Dx + Ey + F = 0. To find the center and radius, group the x-terms and y-terms and complete the square on each group to force the equation back into standard form.

Guided practice

• 1.Write the standard-form equation of a circle with center (0, 0) and radius 5.
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  x² + y² = 25.
• 2.Write the standard-form equation of a circle with center (2, −6) and radius 3.
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  (x − 2)² + (y + 6)² = 9.
• 3.Find the center and radius of the circle (x + 1)² + (y − 5)² = 64.
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  Center (−1, 5), radius 8 (since 64 = 8²).
• 4.Find the center and radius of x² + y² + 4x − 10y + 13 = 0.
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  Group: (x² + 4x) + (y² − 10y) = −13. Complete the square: add (2)²=4 and (−5)²=25 to both sides: (x + 2)² + (y − 5)² = −13 + 4 + 25 = 16. Center (−2, 5), radius 4.
• 5.A circle has equation x² + y² = 20. Is the point (4, 2) on the circle, inside it, or outside it? Explain using the equation.
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  Substitute: 4² + 2² = 16 + 4 = 20, which equals the right side exactly, so (4, 2) is on the circle. If the left side were less than 20 the point would be inside; if greater, outside — because the left side is the squared distance from the center.