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

276 ANALYTIC GEOMETRY
Half the sum of the roots of this equation is X; hence,
by (1),
(6) X=-2.
Since Y is the ordinate of the point on the line (5) whose
abscissa is given by (6), we have
l(?> Xf vmx'.
(7),Y (2 ) 2
We now have three equations, (4), (6), and (7), from which
to eliminate the auxiliary variables x' and y'. We solve (6)
and (7) for x' and y', obtaining
fI mX
x =-2 X, y' =_
Substituting these values of x' and y' in (4) and simplifying
the result, we have
4 Y2= - X.
Consequently, the locus of P is a parabola with vertex at
the origin and opening out along the negative axis of x as
axis. The origin is not a point of the locus.
EXERCISES
1. A variable tangent to the circle
x2 + y2 = a2
meets the hyperbola
2 xy = a2
in the points P1 and P2. Find the equation of the locus of the
mid-point of P1P2. Plot the locus.
Ans. - + 1 4 a curve which does not, despite its apX2 y2 a2
pearance, consist of two conjugate rectangular hyperbolas.
2. Two equal parabolas have the same axis and vertex, but
open in opposite directions. Find the locus of the mid-points
of the chords of one which, when produced, are tangent to the
other.