An equation such as connects two variables: and . We say an ordered pair satisfy the equation if it makes the equation true when the values for and are substituted into the equation. For example, satisfy the given equation. There are infinite number of ordered pairs that satisfy such an equation.
The graph of an equation between and (or any other two variables) is the set of all points satisfying the equation. The phrase “plot the graph of an equation” or “sketch the graph of an equation” means to draw enough number of points and connect them by a smooth curve to illustrate the main features of the graph. Graphs of some equations are sketched in Figure 1.
Circle
Now let’s find the equation of a circle with radius and center . By definition, this circle is the set of all points in a plane that are at a distance from the point (see the following figure). From the Distance formula, we have
By squaring both sides, we obtain:
The
standard form of the equation of a circle with radius and center is

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Figure 2: All points that are at a distance from satisfy the equation |
Example
Write an equation of the circle that has center and contains the point .
Solution
Because is on the circle, the radius is (also denoted by ). By the Distance formula, we have
Substituting , and in Equation
gives us
This circle is illustrated in the following figure.

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Figure 3: Graph of . |
The standard form of the equation of a circle with radius 2 and center is
We can change the look of this equation by expanding the terms on the left hand side (using the formula for the Square of a Sum on this page):
or
Subtracting 4 from both sides, the equation of the circle becomes
The above equation is called the general form of the equation of the circle.
The general form of the equation of a circle is