THE LOCUS OF AN EQUATION
In
chapter 1 of this course, methods for analysis of linear equations are
presented. If a group of x and y values [or ordered pairs, P(x,y)] that satisfy
a given linear equation are plotted on a coordinate system, the resulting graph
is a straight line.
When
higherordered equations such as
are
encountered, the resulting graph is not a straight line. However, the points
whose coordinates satisfy most of the equations in x and y are normally not
scattered in a random field. If the values are plotted, they will seem to
follow a line or curve (or a combination of lines and curves). In many texts
the plot of an equation is called a curve, even when it is a straight line.
This curve is called the locus of the equation. The locus of an equation is a
curve containing those points, and only those points, whose coordinates satisfy
the equation.
At
times the curve may be defined by a set of conditions rather than by an
equation, though an equation may be derived from the given conditions. Then the
curve in question would be the locus of all points that fit the conditions. For
instance a circle may be said to be the locus of all points in a plane that is
a fixed distance from a fixed point. A straight line may be defined as the
locus of all points in a plane equidistant from two fixed
points. The method of expressing a set of conditions in analytical form gives
an equation. Let us draw up a set of conditions and translate them into an
equation.
EXAMPLE: What is the equation of the curve that is the locus
of all points equidistant from the two points (5,3) and (2,1)?
SOLUTION: First, as shown in figure 22, choose some point
having coordinates (x,y). Recall from chapter 1 of this course that the
distance between this point and (2,1) is given by
The distance between point (x,y) and (5,3) is given by
Equating these distances, since the point is to be
equidistant from the two given points, we have
Squaring both sides, we have
Expanding, we have
Canceling and collecting terms, we see that
This is the equation of a straight line with a slope of
minus 1.5 and a y intercept of + 7.25.
Figure 22.Locus of points equidistant from two given
points.
EXAMPLE: Find the equation of the curve that is the locus of
all points equidistant from the line x =  3 and the point (3,0).
SOLUTION. As shown in figure 23, the distance from the
point (x,y) on the curve to the line x = 3 is
The distance from the point (x,y) to the point (3,0) is
Equating the two distances yields
Squaring and expanding both sides yields
Canceling and collecting terms yields
which is the equation of a parabola.
Figure
23.Parabola.
EXAMPLE: What is the equation of the curve that is the locus
of all points in which the ratio of its distance from the point (3,0) to its
distance from the line x = 25/3 is equal to 3/5? Refer to figure 24.
SOLUTION: The distance from the point (x,y) to the point
(3,0) is given by
The distance from the point (x,y) to the line x = 25/3 is
Figure
24.Ellipse.
Since
_{
}
then
Squaring
both sides and expanding, we have
^{
}
Collecting
terms and transposing, we see that
Dividing
both sides by 16, we have
_{
}
This
is the equation of an ellipse.
PRACTICE
PROBLEMS:
Find
the equation of the curve that is the locus of all points equidistant from the
following:
l.
The points (0,0) and (5,4).
2.
The points (3,  2) and (  3,2).
3.
The line x =  4 and the point (3,4).
4.
The point (4,5) and the line y = 5x  4.
HINT:
Use the standard distance formula to find the distance from the point P(x,y)
and the point P(4,5). Then use the formula for finding the distance from a
point to a line, given in chapter 1 of this course, to find the distance from
P(x,y) to the given line. Put the equation of the line in the form Ax + By +
C=O.
ANSWERS:
