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

SPHERES AND OTHER SURFACES

529

means of it the equation of the tangent plane to (6) at the
point (3, - 6, 4) can be written down directly. We have,
namely,
3x - 6y + 4z -(x + 3)+ 3(y - 6)+ 2(z + 4)- 35 = 0,
and this reduces to the equation (7) obtained by the indirect
method.
EXERCISES
Find the equation of the tangent plane to each of the following spheres at the given point.
1. x+y2+z2= 9 at(2,-2,-1).
2. 2 + y2 +z2 = 49 at (3, - 6, 2).
3. (X - 1)2 + (y - 2)2 + (z + 3)2= 81 at (2, 6, 5).
4. x2 +(y + 5)2 + ( - 4)2= 9 at (1, - 3, 2).
5. x2 + y2 + Z2 2 - 4y + 4z = 0 at the origin.
6. x2 + y2 + Z2 - 6 x + 4y + O1z - 11 = 0 at (1, 1, 1).
7. Find the volume of the tetrahedron cut from the first
octant by the tangent plane at (1, 2, 3) to the sphere
2x2 + 2y2 + 2z2+ 2 x -3 y -4z - 12 =0.
8. The coordinates of one of the points of intersection of the
plane 2x-y-2 = 0 with the sphere of Ex. 1 are (2, 2, ).
Find the angle between the plane and the sphere.
Ans. 72~ 39'.
9. Find the angle which the line x = y = z makes with the
sphere of Ex. 5. Ans. 11~ 6'.
5. The Circle. A plane intersects a sphere in a circle, is
tangent to it, or fails to meet it, according as the distance D
of the center of the sphere from the plane is less than, equal
to, or greater than, the radius p of the sphere.
In other words, the equations of a sphere and a plane, considered simultaneously, represent a circle, a point, or no point
whatever, according as D < p, D = p, or D > p.