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

INTRODUCTORY PROBLEMS IN LOCI

87

10. Show that the equation of the circle described on the
line-segment joining the points (x1, Yl), (x2, Y2) as a diameter
may be written in the form
(x - x)(X - X2) + (Y - Y1)(Y - Y2)= 0 -Suggestion. Find the locus of a point P moving so that the
two given points always subtend at P a right angle.
11. The two points, P and P', are symmetric in the line,
x - y = 0, bisecting the angle between the positive axes of x
and y. Show that, if (x, y) are the coordinates of P, then
(y, x) are the coordinates of P'.
12. Prove that a curve is symmetric in the line x - y = 0
if the interchange of x and y in its equation leaves the equation
unchanged.
13. If P and P' are symmetric in the line x +y= 0 and
P has the coordinates (x, y), show that the coordinates of P'
are (- y, - x).
14. Give a test for the symmetry of a curve in the line
x+y =0.
15. Test each of the following curves for symmetry in the
lines x - y = 0 and x + y = 0.
(a) y = a2; (c) x2 - y2 = a2;
(b) xy = - a2; (d) (a - y)2 2x - 2y = 0.
16. Plot the curve of Ex. 15, (d).
In each of the following exercises find the equation of the
locus of the point P. Plot the locus from the equation, making all the use possible of the theory of symmetry.
17. The distance of P from the line x +2 =0 equals its
distance from the point (2, 0).
18. The sum of the distances of P from the points (3, 0)
and (- 3, 0) is 10.
19. The difference of the distances of P from the points
(5, 0) and (-5, 0) is 8.