THE PARABOLA
A parabola is the locus of all points in a plane
equidistant from a fixed point, called the focus, and a fixed line, called the directrix. In the parabola shown in figure 28, point V, which
lies halfway between the focus and the directrix, is called the vertex of the parabola. In this figure and in many of the
parabolas discussed in the first portion of this section, the vertex of the
parabola falls at the origin; however, the vertex of the parabola, like the
center of the circle, can fall at any point in the plane.
The distance from the point (x,y) on the curve to the focus
(a,0) is
The distance from the point (x,y) to the directrix x =  a
is
Since by definition these two distances are equal, we may
set them equal:
Squaring both sides, we have
Figure 28.The parabola.
Expanding, we have
x^{2}2ax+a^{2}+y2=x^{2}+2ax+a^{2}
Canceling and combining terms, we have an equation for the
parabola:
y2 = 4ax
For every positive value of x in the equation of the
parabola, we have two values of y. But when x becomes negative, the values of y
are imaginary. Thus, the curve must be entirely to the right of the Y axis when
the equation is in this form. If the equation is
y2 =  4ax
the curve lies entirely to the left of the Y axis. If the
form of the equation is
x^{2} = 4ay
the curve opens upward and the focus is a point on the Y
axis. For every positive value of y, you will have two values of x, and the
curve will be entirely above the X axis. When the equation is in the form
x^{2} =  4ay
the curve opens downward, is entirely below the X axis,
and has as its focus a point on the negative Y axis. Parabolas that are
representative of the four cases given here are shown in figure 29.
When x is equal to a in the equation
y2 = 4ax
then
y2 = 4a2
and
y=2a
This value of y is the height of the curve at the focus or
the distance from the focus to point D in figure 28. The width of the curve at
the focus, which is the distance from point D to point D' in
Figure 29.Parabolas corresponding to four forms of the
equation.
the figure, is equal to 4a. This width is called the focal
chord. The focal chord is one of the properties of a
parabola used in the analysis of a parabola or in the sketching of a parabola.
EXAMPLE:
Give the length of a; the length of the focal
chord; and the equation of the parabola, which is the locus of all points
equidistant from the point (3,0) and the line x =  3.
220
Figure 210.Sketch of a parabola.
SOL UTION: First plot the given information
on a coordinate system as shown in figure 210, view A. Figure 28 shows you
that the point (3,0) corresponds to the position of the focus and that the line
x =  3 is the directrix of the parabola. Figure 28 also shows you that the
length of a is equal to one half the distance from the focus to the directrix
or, in this problem, one half the distance from x =  3 to x = 3. Thus, the
length of ais3.
The second value required by the problem is the length of
the focal chord. As stated previously, the focal chord length is equal to 4a.
The length of a was found to be 3, so the length of the focal chord is 12.
Figure 28 shows that one extremity of the focal chord is a point on the curve
2a or 6 units above the focus, and the other extremity is a second point 2a or
6 units below the focus. Using this information and recalling that the vertex
is onehalf the distance from the focus to the directrix, plot three more
points as shown in figure 210, view B.
Now a smooth curve through the vertex and the two points
that are the extremities of the focal chord provide a sketch of the parabola in
this problem. (See fig. 210, view C.)
To find the equation of the parabola, refer to figure
210, view D, and use the procedure used earlier. We know by definition that
any point P(x,y) on the parabola is equidistant from the focus and directrix.
Thus, we equate these two distances:
However, we have found distance a to be equal to 3, so we
substitute:
We square both sides:
Then we expand:
We cancel and combine terms to obtain the equation of the
parabola:
If we check the consistency of our findings, we see that
the form of the equation and the sketch agree with figure 29, view A. Also,
the 12 in the right side of the equation corresponds to the 4a in the standard
form, which is correct since we determined that the value of a was 3. Or, since
the curve is entirely to the right of the Y axis, then we can apply the formula
y^{2} = 4ax by substituting a = 3 to give
NOTE:
When the focus of a parabola lies on the Y axis, the equated distance equation
is
