Calculus: Analytic Geometry and Calculus, with Vectors

1.4 Distances, circles, and parabolas^27

is often printed instead of the built-up quotient bc Learning to read

printed mathematics involving shilling quotients is an art that must be
cultivated, and this is a good opportunity. Since "multiplication takes
precedence over division" the quotient a/bc means a/(bc) and does not

mean (alb)c. Thus, for example, 1/2k means 1/(2k) or

1
2kand does not
mean (1/2)k or 4k. When the next paragraph is read, the quotients
should be handwritten in built-up forms so the calculations can be made
more easily. If troubles appear, the difficulty may be the canonical one
that arises when a printer converts an author's 1/2k into 4k. Every-
thing should be checked.
We can get experience with the distance formula by starting to learn
about parabolas. A parabola is, as we shall show in Section 6.2, the set
of points (in a plane) equidistant from a fixed point F which is called the
focus and a fixed line L which is called the dtrectrix and which does not
pass through the focus.t In order to obtain the equation of a parabola in
an attractive form, we let 1/2k denote the distance from F to L so that
1/2k = p and k = 1/2p, where p is the distance (length of the "per-
pendicular") from F to L. Then we put the y axis through F perpen-
dicular to L and put the x axis midway between F and L as in Figure 1.47.

x

Figure 1.47

The parabola is the set of points P(x,y) for which FP = DP. Using the

distance formula and the fact that y + 1/4k > 0 when FP = DP gives

(1.471)

1

X

2

+ \y -4k)2-y + 4k

t The assumption that F is a "fixed" point and L is a "fixed" line means merely that F

and L are "given" or "selected" in some way. There is no implication that other points and

lines are "unfixed" in the sense that they are moving. At one time the parabola was

defined as the path (or locus) of a point P which moves in such a way that it is always equi-

distant from F and L. There are reasons why it is better to say that a parabola is a point

set. Everybody knows that pencil points and numerous other things move, but even if we

swallow the dubious idea that "mathematical points" can move we still find that the old-

fashioned definition does not tell how a point P should move to trace the whole parabola and

not merely a part of the parabola.