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

520

ANALYTIC GEOMETRY

6. Do Ex. 17 at the end of Ch. XX without finding the
coordinates of the point of intersection of the given line and the
given plane.
7. Find the equations of the line which contains the point
(2, O, - 1) and intersects each of the lines
J2x —y+3z=0, f +y+2z-5=0,
33x+y-2z=2; 3x+4y-z+1=0.
8. Find the equations of the line which intersects each of the
lines given in Ex. 7 and is parallel to the line 4- 6 x = y + 9 = 2 z.
9. A plane intersects the (x, y)-plane in the line whose equation in the (x, y)-plane is 2 x + 3y = 12. If the plane cuts from
the first octant a tetrahedron whose volume is 12, find its
equation. Ans. 2x + 3y+ 4z- 12 =0.
10. There are two planes which contain the line
x+2y+z++1 =0, 2x +y- z-7=0
and make angles of 30~ with the plane x - z + 2 = 0. Find
their equations. Ans. x- y - 2z-8=0, 2x+y -z-7=0.
11. Find the equations of the planes which contain the line
given in Ex. 10 and are /2 units distant from the point
(2, 2, - 3).
12. The planes through the edges of a trihedral angle perpendicular to the opposite faces pass through a line. Prove this
theorem in the case that the faces lie in the planes
2x -y+z=0, 4x —y+3z=0, 3x-2y —z=0.
13. Prove the theorem of Ex. 12 in the general case.
THE EQUATIONS Xu +,uv = 0, uv = 0 *
14. THEOREM. If u = 0, v = 0 are the equations of two surfaces, the equation Xu+tv =0, X/A = 0, represents in generalt a
*Cf. Ch. IX, ~~ 3, 4.
t In particular, it may represent a curve or a point; cf. footnotes, pp.
445, 167.