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

ADVANCED METHODS

521

surface which contains the total intersection of the two surfaces,
if they intersect, and has no other point in common with either of
them. If the given surfaces do not intersect, the equation represents in general * a surface not meeting either of them or it has no
locus. Prove this theorem.
15. Find the equation of the sphere which contains the
circle
2 + y2 + -2 _ 4 = 01 z- 5 = 0
and passes through the point (3, 0, 2).
Ans. 2 + y2 + 2+ 3 z= 19.
16. Prove that the curve of intersection of the cylinders
X2 + y2 = 4, y2 + z2 = 5 lies on the surface x2 - z2 + 1 = 0 and is
the total intersection of this surface with each cylinder.
17. THEOREM. If u = 0, v = 0, w = 0 are the equations of
three planes which meet in a single point, the equation
Xu ~- tv + vw = 0 represents a plane through this point. Conversely, every plane which contains this point is a linear combination of the three given planes. Prove this theorem.
18. Find the equation of the plane which is determined by
the points (0, 0, 0), (1, 2, 0) and the point of intersection of the
three planes
2x+y+3z+1=0, x+y-4z+2=0, 2x-2y+5z-3=0.
Ans. 2x-y-6z=0.
19. THEOREM. The equation uv = 0 represents those points
and only those points which lie on the surfaces u = 0 and v = 0.
Prove this theorem.
20. What do the following equations represent?
(a) 2 - y2 = 0; (c) x2- y- x + y = 0;
(b) 4- y4 = 0; (d) xy -xz -2y+2z =.
* In particular, it may represent a curve or a point; cf. footnotes, pp.
445, 168.