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

528

ANALYTIC GEOMETRY

Since the point (xl, Yl, z1) lies on the sphere (1),
X12 + yi2 + z12 _ p2
and hence the equation of the tangent plane, in final form, is
(2) xIx + yly + zz = p2.
In a similar manner the equation of the tangent plane to
the sphere
(3) (X - a)2 + (y _ -)2 + (Z _ y) = p2
at the point (x1, yl, z1) of the sphere cal be shown to be
(4) (xI - )(x - )+(y - P)(y - )+ (Z - y)(z- y) = p2.
The use of (4) to find the equation of the tangent plane to
a sphere whose equation is in the form
(5) x2 + y2 + z2 + Ax + By + Cz + D = 0
involves the reduction of the equation of the sphere to the
form (3). Thus, if the sphere is
(6) 2 + y2+2- 2 x + 6y + 4z - 35 = 0,
the equation must first be rewritten as
(x - l)2 + (y +3)2 + (z + 2)2= 49.
The equation of the tangent plane at the point (3, - 6, 4), for
example, is then, according to (4),
(3 - 1)(x - 1) + (- 6 + 3)(y + 3) + (4 + 2)(z + 2) =49,
or
(7) 2- 3y + 6z- 48=0.
The coordinates of the center (a, f, y) and the square of the
radius, p2, of a sphere whose equation is in the form (5) are
given by formulas (4), ~ 2. If these values for a, fi, y, p2 are
substituted in (4) and the equation obtained is simplified, the
result is
(8) xx + yly + z1z + (x + x1)+ ((y +y+ )+j(z + + D=O.
This is the equation of the tangent plane at the point
(x1, Y1, z1) to a sphere whose equation is in the form (5). By