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

A SECOND CHAPTER ON LOCI

273

same side of AB and relatively so situated that OR. OS = b2
is then given by the equation,
(1) rs = b2.
The equations of AR and BS are
x-+___, y + l.
X I- "= ~, x+~=1.
-aSince P: (X, ) is the point of intersection of these lines, we
Since P' (X, Y) is the point of intersection of these lines, we
have
(2) X Y
a r
(3) 1 X.
a s
To eliminate the auxiliary variables r and s from equations
(1), (2), (3) is now our problem. We notice that in the product
of equations (2) and (3):
X2 y2
a2 rs
r and s enter only in the form rs, and that the value of rs is
given by (1) as b2. We have, therefore, as the equation of the
locus
X2 ye
a2 b2
The locus of P is, therefore, an ellipse, with its axes along
AB and L, and passing through the points A and B. These
points are not, however, points of the locus.
EXERCISES
1. What is the locus of P in the problem in the text, if R
and S are always on opposite sides of AB?
2. The points P1 and P2 are fixed, and the lines L1 and L2
are perpendicular to PiP2 in P1 and P2 respectively; Qi and
Q2 are two points on L1 and L2 respectively, both on the same
side of P1P2 and moving so that the product of their distances