Read this section and work through practice problems 1-9.
If we move from a point to a point in the plane, then we will have two increments or changes to consider. The increment in the or horizontal direction is which is denoted
by . The increment in the or vertical direction is . These increments are shown in Fig. 5 . does not represent times , it represents the difference in
the coordinates: .
The distance between the points and is simply an application of the Pythagorean formula for right triangles, and
The midpoint of the line segment joining and is
Example 2: Find an equation describing the points which are equidistant from and . (Fig. 6)
Solution: The points must satisfy so
By squaring each side we get
Then
so and , a straight line. Every point on the line is equally distant from and .
Practice 2: Find an equation describing all
points equidistant from and .
A circle with radius and center at the point consists of all points which
are at a distance of from the center : the points which satisfy .
Example 3: Find the equation of a circle with radius and center
. (Fig. 7)
Solution: A circle is the set of points which are at a fixed distance from the center point , so this circle will be the set of points
which are at a distance of 4 units from the point . P will be on this circle if .
Using the distance formula and simplifying,
.
Practice 3: Find the equation of a circle with radius and center .