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

THE PLANE

467

19. There are two planes parallel to the plane 2 x- 6 y+3
= 4 and distant 3 units from the origin. Find their
equations.
20. Find the equation of the plane parallel to the plane
given in Ex. 19 and so located that the point (3, 2, 8) is midway between the two planes. Ans. 2 x- 6 y + 3 z - 32 = 0.
21. Three faces of a box lie in the planes 2x —y= 6,
x + 2 y = 8, z = 8 and a vertex is at the point (9, 5, 2). Find
the equations of the planes of the other three faces.
22. Find the equation of the plane which passes through
the point (2, - 1, 8) and is parallel to each of two lines having 2, - 3, 4 and 5, - 7, 8 as their direction components.
Ans. 4x+4y+z — 12 =0.
23. Show that the equation of the plane passing through
the points (xj, yi, z2), (x2, Y2, z2) and perpendicular to the plane
Ax + By + Cz + D = 0 can be written in the form
x y z 1
xi Y1 z1 1
x2 Y2 2 1
A B C 0
/24. Show that the four planes,
2x-y- -3= =, x- y+2z -3= 0,
x -2 y + - 0, x +- y z- 6=0,
meet in a point.
25. The six planes, each of which passes through the midpoint of an edge of a tetrahedron and is perpendicular to the
opposite edge, go through a point.
Prove this theorem for the tetrahedron of Ex. 9, ~ 6.
26. Prove the theorem of Ex. 25 for the general tetrahedron,
choosing the coordinate axes skillfully.
27. Show that the plane
2x + Y3 y 4 z = 5
ff"T ~~yT 7i