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

598

ANALYTIC GEOMETRY

remove the terms in xy, yz, zx and then a proper change of
origin, as suggested by the new equation, is made.
EXERCISES
1. Find the equations of the rotation of the axes which
introduces the two directed lines through 0 with the direction
cosines -,,- 3 and, 3, - as the axes of x' and y' respectively.
2. Find the equations of a rotation of the axes which introduces the planes,
x+2y+2z=0, 2x-2y+z=0, 2x+y-2z=0,
as the (x', y')-, (y', z')-, and (z, x')- planes, respectively.
3. Find the equations of a rotation of the axes which introduces the planes
3x+ 4y+12z= 0, 12x+3y-4z==0
as the (x', y')- and (y', z')- planes.
4. Transform the equation of the hyperbolic paraboloid
2 - y2 = 2 mz
by a rotation of the x- and y-axes through an angle of - 45~
about the axis of z. Ans. x'y' = m z'.
Determine the surface represented by each of the following
equations.
5. 5x2+5y2+4z2- 6xy-32=0.
6. 3x2-2y2-_5z2-4yz+4y+10z+1=0.
7. x2-2y2+z2_-2xz-x+3z-3=0.
8. Transform the equation of the surface
5 x2 -2 y2 + 11z2 + 12 xy +12 yz -14 = 0
by the rotation of the axes of Ex. 1. Thus identify the surface.
9. The equation
10x2 + 13By2 +13 2 - 4xy -10 yz - 4xz- 36 y + 36 z = 0