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

286

ANALYTIC GEOMETRY

19. Determine the locus of the mid-point of a variable
chord of an ellipse drawn from a vertex.
20. Find the locus of the mid-point of a variable chord of
a parabola which subtends a right angle at the vertex.
21. A point R traces an ellipse, of which A and A' are the
vertices. Find the locus of the point of intersection of the
lines drawn through A and A' perpendicular to AR and A'R
respectively.
Ans. An ellipse, similar to and with the same center as the
given ellipse, but with opposite transverse and conjugate axes.
22. The asymptotes of a hyperbola intercept the segment
AB on a variable tangent. What is the locus of the point
dividing AB in a given ratio, mi: n2?
Ans. A similar hyperbola, with the same transverse and
conjugate axes.
Exercises 23-25. Determine the equations of the desired loci
by use of rectangular coordinates. To identify the locus from
its equation introduce polar coordinates.
23. Find the locus of the point of intersection of a variable
tangent to a rectangular hyperbola with the line through the
center perpendicular to the tangent. Ans. A lemniscate.
24. Find the locus of the point of intersection of a variable
tangent to the circle x2 + y2 + 2 ax = 0 and the perpendicular
to this tangent from the origin. Ans. A cardioid.
25. What is the locus of the mid-points of the chords of the
circle x2 + y2 = a2 which, when produced, are tangent to the
hyperbola 2 xy = c2? Ans. A lemniscate.
26. Show that the line
x cos 30~ + y sin 30~ = 5
is 5 units distant from the origin and that the perpendicular
from the origin to it makes with the positive axis of x an angle
of 30~. Prove that the distance of the point (xo, yo) from the
line is
- (xo cos 30~ + yo sin 30~ - 5),