Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

CHAPTER VI

THE PARABOLA
1. Definition. A parabola is defined as the locus of a point
P, whose distance from a fixed line D is always equal to its
distance from a fixed point F, not on
D ~ the line. It is understood, of course,
-— ~ ~ that P is restricted to the plane determined by D and F.
One point of the locus is the mid-point
F 4A, Fig. 2, of the perpendicular FE
dropped from F on D. Through A
draw T parallel to D. Then no other
point on T, or to the left of T, can belong
FIG. 1 to the locus, for all such points are
clearly nearer to D than they are to F.
Further points of the locus can be
obtained as follows. To the right of T
draw L parallel to D, cutting AF, pro- D T:
duced if necessary, in S. With ES as
radius and F as center describe a circle-,
cutting L in P and Q. Then P and Q.
lie on the locus. A S
A large number of points having been
obtained in this way, a smooth curve
can be passed through them. The curve
is symmetric in the line AF, and evi-G. 2
dently has T as a tangent.
The line D is called the directrix, and the point F, the focus,
of the parabola; A is the vertex, and the indefinite line AF,
the axis; FP is a focal radius.

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