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

DIAMETERS. POLES AND POLARS 311
EXERCISES
1. Four points P1, P2, Q1, Q2 on the axis of x have, respectively, the abscissas 3, 8, 5, - 7. Show that Q1, Q2 divide PiP2
harmonically, and find the common ratio pu of internal and external division. Find, also, the value of the ratio, L', for the
division by P1 and P2 of the segment Q1Q2.
2. Find the point on the axis of x which, with the point
(- 1, 0), divides harmonically the segment of the axis joining
the points (- 8, 0), (3, 0).
3. Exercise 1, for the four points P1, P2, Q1, Q2 with the
respective coordinates (2, 3), (- 1, 9), (1, 5), (5, - 3).
4. Find the point which, with the point (2, 1), divides harmonically the line-segment joining the points (5, - 2), (1, 2).
9. Polar of a Point. Consider the following locus problem.
The ellipse y
) P:y2 XY)
a2 b2 Q -:l (x XI
and the point PI: (xl, yl) are -
given. A line L is drawn through
P1 meeting the ellipse in Q, and
Q2, and on L the point P: (X, Y) FIG. 20
is marked which, with PI, divides
QiQ2 harmonically. What is the locus of P, as L revolves
about P?
Since P1, P divide QlQ2 harmonically, Q1, Q2 divide P1P
harmonically. Hence the coordinates (1', Y1/), (x2', Y2) of Q1,
Q2 are:
Q1: _ + b Yl + fit Y
Qx1 x/= Y, Y - -,, x _ - /-X = p Y1.-K Y
Q2: X2 = y l -2 _ _,_ Y2 -1 —/A 1 —/
As L rotates, the ratio f varies; it is, then, an auxiliary
variable expressing analytically the rotation of L.