Equations of Circles
A circle can be defined as the set of all points in a plane that are equidistant from a
fixed point. The fixed point is the center of the circle, and the distance between the
center and a point on the circle is the radius (see the figure below).
You can use the Distance Formula to write an equation for the circle with center
(h, k)
and radius r. Let (x, y) be any point on the circle. Then the distance between
(x, y)
and the center (h, k) is given by
By squaring both sides of this equation, you obtain the standard form of the
equation of a circle.
Standard Form of the Equation of a Circle
The point (x, y) lies on the circle of radius r and center (h, k) if and only
if (x  h)^{2} + (y  k)^{2} = r^{2}.
The standard form of the equation of a circle with center at the origin,
(h, k) = (0, 0), is x^{2} + y^{2} = r^{2}.
If r = 1, the circle is called the unit circle.
Example 1
Finding the Equation of a Circle
The point (3, 4) lies on a circle whose center is at (1, 2), as shown in the
figure below. Find an equation for the circle.
Solution
The radius of the circle is the distance between (1, 2) and (3, 4).
You can write the standard form of the equation of this circle as
Standard form
By squaring and simplifying, the equation (x  h)^{2} + (y  k)^{2}
= r^{2 }can be
written in the following general form of the equation of a circle.
Ax^{2} + Ay^{2} + Dx + Ey + F = 0, A
≠ 0
To convert such an equation to the standard form
(x  h)^{2} + (y  k)^{2} = p
you can use a process called completing the square. If p > 0, the graph of the
equation is a circle. If p = 0, the graph is the single point (h, k). If p < 0, the
equation has no graph.
Example 2
Completing the Square
Sketch the graph of the circle whose general equation is
4x^{2} + 4y^{2} + 20x  16y + 37 = 0
Solution
To complete the square, first divide by 4 so that the coefficients of x^{2}
and y^{2}
are both 1.
4x^{2} + 4y^{2} + 20x  16y
+ 37 
= 0 
General form 
x^{2} + y^{2} + 5x  4y

= 0 
Divide by 4.



Group terms. 


Complete the square by adding
and 4 to both sides. 

= 1 
Standard form 
Note that you complete the square by adding the square of half the coefficient of x and
the square of half the coefficient of y to both sides of the equation. The circle is
centered at
and its radius is 1, as shown in the figure below.
