**Equation Of Circle**

The general form for the equation of a *circle* in the *xy* plane is given by the following formula:

( *x* − *x* _{0} ) ^{2} + ( *y* − *y* _{0} ) ^{2} = *r* ^{2}

where ( *x* _{0} , *y* _{0} ) represents the coordinates of the center of the circle, and *r* represents the *radius* . This is illustrated in Fig. 3-8. In the special case where the circle is centered at the origin, the formula becomes

*x* ^{2} + *y* ^{2} = *r* ^{2}

Such a circle intersects the *x* axis at the points ( *r* , 0) and ( *−r* , 0); it intersects the *y* axis at the points (0, *r* ) and (0, − *r* ). An even more specific case is the *unit circle* :

*x* ^{2} + *y* ^{2} = 1

This curve intersects the *x* axis at the points (1, 0) and (−1, 0); it also intersects the *y* axis at the points (0, 1) and (0, −1).

**Graphic Solution To Pairs Of Equations**

The solutions of pairs of equations can be found by graphing both the equations on the same set of coordinates. Solutions appear as intersection points between the plots.

**Graphic Solution To Pairs Of Equations Practice Problems**

**Example 1 **

Suppose that you are given these two equations and are told to solve them for values of *x* and y that satisfy both at the same time:

*y* = *x* ^{2} + 2 *x* + 1

*y* = *−x* + 1

These equations are graphed in Fig. 3-9. The line crosses the parabola at two points, indicating that there are two solutions to this set of simultaneous equations. The coordinates of the points corresponding to the solutions are

( *x* _{1} , *y* _{1} ) = (−3, 4)

( *x* _{2} , *y* _{2} ) = (0, 1)

**Example 2 **

Here is another pair of “two by two” equations (two equations in two variables) that can be solved by graphing:

*y* = −2 *x* ^{2} + 4 *x* − 5

*y* = −2 *x* − 5

These equations are graphed in Fig. 3-10. The line crosses the parabola at two points, indicating that there are two solutions. The coordinates of the points corresponding to the solutions are

( *x* _{1} , *y* _{1} ) = (3, −11)

( *x* _{2} , *y* _{2} ) = (0, −5)

Sometimes a graph will reveal that a pair of equations has more than two solutions, or only one solution, or no solutions at all. Solutions to pairs of equations always show up as intersection points on their graphs. If there are *n* intersection points between the curves representing two equations, then there are *n* solutions to the pair of simultaneous equations. However, graphing is only good for estimating the values of the solutions. If possible, algebra should be used to find exact solutions to problems of this kind. If the equations are complicated, or if the graphs are the results of experiments, it will be difficult to use algebra to solve them. Then graphs, with the aid of computer programs to accurately locate the points of intersection between graphs, offer a better means of solving pairs of equations.

Practice problems for these concepts can be found at: Graphing Schemes for Physics Practice Test

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