220 4 Geometry and Trigonometry
- x^2 −y^2 −z^2 =1, hyperboloid of two sheets;
- x^2 +y^2 =z, elliptic paraboloid;
- x^2 −y^2 =z, hyperbolic paraboloid.
In Cartesian coordinates, in these formulas there is a scaling factor in front of each
term. For example, the standard form of an ellipsoid in Cartesian coordinates is
x^2
a^2+
y^2
b^2+
z^2
c^2= 1.
As in the case of conics, the equation of the tangent plane to a quadric at a point
(x 0 ,y 0 ,z 0 )is obtained by replacing in the equation of the quadricx^2 ,y^2 , andz^2 , respec-
tively, byxx 0 ,yy 0 , andzz 0 ;xy,xz, andyz, respectively, byxy^0 + 2 yx^0 ,xz^0 + 2 zx^0 , andyz^0 + 2 zy^0 ;
andx,y, andzin the linear terms, respectively, byx+ 2 x^0 ,y+ 2 y^0 , andz+ 2 z^0.
Our first example comes from the 6th W.L. Putnam Mathematical Competition.
Example.Find the smallest volume bounded by the coordinate planes and by a tangent
plane to the ellipsoid
x^2
a^2+
y^2
b^2+
z^2
c^2= 1.
Solution.The tangent plane to the ellipsoid at(x 0 ,y 0 ,z 0 )has the equation
xx 0
a^2+
yy 0
b^2+
zz 0
c^2= 1.
Itsx,y, andzintercepts are, respectively,a
2
x 0 ,
b^2
y 0 , andc^2
z 0. The volume of the solid cut off
by the tangent plane and the coordinate planes is therefore
V=
1
6
∣
∣∣
∣
a^2 b^2 c^2
x 0 y 0 z 0∣
∣∣
∣.
We want to minimize this with the constraint that(x 0 ,y 0 ,z 0 )lie on the ellipsoid. This
amounts to maximizing the functionf (x, y, z)=xyzwith the constraint
g(x, y, z)=
x^2
a^2+
y^2
b^2+
z^2
c^2= 1.
Because the ellipsoid is a closed bounded set,f has a maximum and a minimum on
it. The maximum is positive, and the minimum is negative. The method of Lagrange
multipliers yields the following system of equations in the unknownsx,y,z, andλ: