An equation such as y=x22+1 connects two variables: x and y. We say an ordered pair satisfy the equation if it makes the equation true when the values for x and y are substituted into the equation. For example, (0,1),(1,1.5),(1,1.5),(2.1,3.205), satisfy the given equation. There are infinite number of ordered pairs that satisfy such an equation.

The graph of an equation between x and y (or any other two variables) is the set of all points (x,y) 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.

(a) Graph of y=12x2+1 (b) Graph of y=34x1
(c) Graph of y2=x

Figure 1

Circle

Now let’s find the equation of a circle with radius r and center C(a,b). By definition, this circle is the set of all points P(x,y) in a plane that are at a distance R0 from the point C(a,b) (see the following figure). From the Distance formula, we have

|PC|=(xa)2+(yb)2=R.

By squaring both sides, we obtain:

(xa)2+(yb)2=R2

The standard form of the equation of a circle with radius R and center (a,b) is

(xa)2+(yb)2=R2

Figure 2: All points P(x,y) that are at a distance R from C(a,b) satisfy the equation (xa)2+(yb)2=R2.

Example

Write an equation of the circle that has center C(1,1) and contains the point P(4,5).

Solution

Because P is on the circle, the radius r is |PC| (also denoted by d(P,C)). By the Distance formula, we have

R=|PC|=(4(1))2+(51)2=25=5.

Substituting a=1,b=2, and r=5 in Equation

gives us

(x+1)2+(y1)2=25.

This circle is illustrated in the following figure.

Figure 3: Graph of (x+1)2+(y1)2=25.

The standard form of the equation of a circle with radius 2 and center (2,3) is

(x2)2+(y+3)2=4.

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):

x24x+4+y2+6y+9=4

or

x2+y24x+6x+13=4

Subtracting 4 from both sides, the equation of the circle becomes

x2+y24x+6x+9=0.

The above equation is called the general form of the equation of the circle.

The general form of the equation of a circle is

x2+y2+Ax+Bx+E=0.