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

86

ANALYTIC GEOMETRY

In each of the following exercises test the given curve for
symmetry in each axis and in the origin. Plot the curve.
3. xy+ =0. 6. y2+4x=O.
4. 10y=x4. 7. x2-y2=4.
5. 20x=y5. S. x2+2y2=16.
EXERCISES ON CHAPTER V
1. The base of a triangle is fixed and the ratio of the
lengths of the two sides is constant. Find the locus of the
vertex. Ans. A circle, except for one value of the constant.
2. A point P moves so that its distance from a given line
L is proportional to the square of its distance to a given point
K, not on L. If P remains always on the same side of L as
K, show that its locus is a circle.
3. Find the locus of P in the preceding exercise, if it remains always on the opposite side of L from K. Does your
answer cover all cases?
4. If, in Ex. 2, K lies on L and P may be on either side of
L, what is the locus of P?
5. Three vertices of a quadrilateral are fixed. Find the
locus of the fourth, if the area of the quadrilateral is constant.
6. Find the locus of a point moving so that the sum of
the squares of its distances from the sides of an equilateral
triangle is constant. Discuss all cases.
Ans. A circle, center at the point of intersection of the medians; this point; or no locus.
7. The feet of the perpendiculars from the point P: (X, Y)
on the sides of the triangle with vertices in the points (0, 0),
(3, 0), (0, 1) lie on a line. Find the locus of P.
Ans. The circle circumscribing the triangle.
8. The preceding problem, if the triangle has the points
(2, 0), (- 3, 0), (0, 4) as vertices.
9. Problem 7, for the general triangle, with vertices at
(a, 0), (b, 0), (0, c).