The standard equation of a circle, the general form, completing the square to find the center and radius, and writing equations from given information.
From distance to equation
A circle with center \((h, k)\) and radius \(r\) is the set of points \((x, y)\) at distance \(r\) from the center. By the distance formula, \(\sqrt{(x - h)^2 + (y - k)^2} = r\). Square both sides:
General form and completing the square
Expanding standard form gives the general form
\[x^2 + y^2 + Dx + Ey + F = 0\]
To get the center and radius back, complete the square in \(x\) and in \(y\).
Writing equations from information
You need the center and the radius.
Practice
Quick check
What is the center of \((x - 3)^2 + (y + 1)^2 = 16\)?
The coordinates have the opposite sign of the numbers in the parentheses.
Quick check
What is the radius of \(x^2 + y^2 = 20\)?
\(r^2 = 20\), so \(r = \sqrt{20} = 2\sqrt{5}\).
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