Calculus: Analytic Geometry and Calculus, with Vectors

392 Cones and conics

Multiplying (5) and (6) by (1 - X) and X, respectively, and adding give the

relation
(9) .4x1x + Byiy = 1,
which shows that the intersection of the tangents lies on the polar line L1 having
the equation (2). This is a remarkable geometric fact. Different lines through
P1 yield hordes of pairs of tangents, and all of the intersections lie on the same
polar line L1. We shall not undertake
to prove the fact, illustrated by Figure
6.4993, that pairs of chords as well as pairs
of tangents intersect on the polar line L1.
p 3 When the lines L and L' through P, in-
tersect the conic at four known points
L1 P2, Ps, P2, P3 as in the figure, the chords

Figure 6.4993

P2P3 and P3P3 intersect at one point on L,,

the chords P2P'3 and P3P2 intersect at

another point on L1, and, moreover, these

two intersections determine Li. Thus

(unless parallelism causes trouble) we can

start with just four points on a conic and

use them to determine a point P1 and its polar line L1. This "four-point con-

struction" is remarkable because the four points on the conic do not determine the

conic. If we are given a fifth point E on the conic (such that no three of the five

are collinear) then the conic is determined and we can produce the dotted lines of

the figure to construct a sixth point E'. Appropriate use of polar points and

polar lines provides methods for construction of more points. We are not

expected to learn much about these matters in our elementary course, but we can

be aware of the fact that many persons continue study of geometry to learn more

6.5 Translation and rotation of axes

advantages gained by introducing supplementary coordinate systems, we

the graph of the equation

x2 -{- y2 - 10x - 8y -}- 32 = 0,

we complete squares and write the equa-

tion in the form

x (6.51) (x-5)2+(y-4)2=9.

Figure 6.52 It is then easily seen th hat t e graph is

a circle with center at (5,4) and radius 3,

and we can sketch the graph in Figure 6.52 without onerous calculations.

If in the equation (6.51) we set

look at an example so simple that the sup-

plementary coordinate system is not

P(x',y') needed. When we want to learn about

(6.511) x' = x - 5, y'=y-4