College Algebra 9th txtbk

126 CHAPTER 2 GRAPHS
We equate the corresponding coordinates and solve the resulting equations for x and y:
1 3 x
2
5 x 3 y
1 1 y
2
2 3 x 2 1 y
*
2 3 3 x 3 2 1 1 y 1
Therefore, B (5, 3).
MATCHED PROBLEM 4
If M (1, 1) is the midpoint of the line segment joining A (1, 5) and B (x, y),
find the coordinates of B.
Z Circles
The distance formula would be helpful if its only use were to find actual distances between
points, such as in Example 2. However, its more important use is in finding equations of
figures in a rectangular coordinate system. We start with an example.
EXAMPLE 5 Equations and Graphs of Circles
Write an equation for the set of all points that are 5 units from the origin. Graph your equation.
SOLUTION
The distance between a point (x, y) and the origin is
d 2(x 0) 2 ( y 0) 2 2x 2 y 2
So, an equation for the set of points that are 5 units from the origin is
2x 2 y 2 5
We square both sides of this equation to obtain an equation that does not contain any radicals.
x 2 y 2 25
Because (x) 2 x 2 and (y) 2 y 2 , the graph will be symmetric with respect to the x axis,
y axis, and origin. We make up a table of solutions, sketch the curve in the first quadrant,
and use symmetry properties to produce a familiar geometric object—a circle (Fig. 4).
x y
0 5
3 4
4 3
5 0
(3, 4)
(4, 3)
(5, 0)
(4, 3)
(3, 4)
y
(0, 5)
(3, 4)
(4, 3)
(5, 0)
x
(4, 3)
(3, 4)
(0, 5)
Z Figure 4
MATCHED PROBLEM 5
Write an equation for the set of all points that are three units from the origin. Graph your
equation.
*Throughout the book, dashed boxes—called think boxes—are used to represent steps that may be performed
mentally.