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

A SECOND CHAPTER ON LOCI

275

The coordinates (ax, yi) and (x2, Y2), of P1 and P2, are the

simultaneous solutions of equations
in (3) the value of y from (2) and
collecting terms, we have the quadratic equation,
19 x2 -16x + 1 = 0,

(2) and (3). Substituting

y
~ (Xy,,Yj)

I

for the determination of xl and x2.
We are interested, not in the
actual values of x1 and x2, but in
half their sum; for this is the
abscissa of the mid-point, P, of
P1P2.- By (1) the sum is 16. Then the abscii
and, since P lies on the line (2), its ordinate is

P
2:(X2,Y2)
FIG. 6
ssa of P is 8;
I T

y = 2(8%-)-1 =-1 9 '
Consider now the following locus problem: A variable tangent to the parabola,
Il. yy2= 2mx,

meets the hyperbola,
xy = 2,
in the points PI and P2.
What is the locus of the midpoint, P, of PiP2?
As auxiliary variables we
take the coordinates (x', y') of
the point R tracing the parabola; they satisfy the equation of the parabola:

FIG. 7

(4)

y2 = 2 mx'.

The tangent at R has the equation
(5) y'y = m( + x').
To find the coordinates of P1 and P2, we solve (5) simultaneously with the equation of the hyperbola. Eliminating y, we
have:
mx1 + rzx'x - 2y' = 0.