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

A SECOND CHAPTER ON LOCI

263

2. One Auxiliary Variable. In illustrating the method by
examples, let us first complete the problem of the previous
paragraph.
Consider P as the point of intersection of VC and BD.
The equation of VC is
(1) X=y.
The slope of AVis
h
y+a
hence the equation of the perpendicular, BD, to AV is
(2) -0=- + a ( -a).
h
Solving equations (1) and (2) simultaneously, we obtain the
coordinates of P,
(3) x= 7=,a2 - y
h
in terms of the constants a and h and the auxiliary variable y.
By eliminating y from equations (3), we obtain
(4) X2 =-h Y+a2
as the equation of the locus.
The locus of P is, then, a parabola, with its axis along the
perpendicular bisector of the base of the triangle; it goes
through the extremities of the base and opens away from the
line L. Every point of it is included in the locus.*
- In the locus problems considered hitherto, particularly in Ch. V,
care was taken to emphasize that two things are necessary: (a) to
determine the curve, or curves, on which points of the locus lie; (b) to
show, conversely, that every point lying on the curve, or curves, obtained
is a point of the locus. In the problems of the present chapter, - for
example, in the one above, - part (b) of the proof is usually omitted. It
consists, as a rule, in retracing the steps of part (a) and so presents, in
general, no difficulty. And it is more important, now, that the student
gain facility in deducing the equation of the curve, or the equations of the
curves, which turn out, in the great majority of cases, to be precisely the
locus.