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Putnam
Putnam 1938
B6
B6
Part of
Putnam 1938
Problems
(1)
Putnam 1938 B6
Source:
8/20/2021
What is the shortest distance between the plane
A
x
+
B
y
+
C
z
+
1
=
0
Ax + By + Cz + 1 = 0
A
x
+
B
y
+
C
z
+
1
=
0
and the ellipsoid
x
2
a
2
+
y
2
b
2
+
z
2
c
2
=
1.
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.
a
2
x
2
+
b
2
y
2
+
c
2
z
2
=
1.
You may find it convenient to use the notation
h
=
(
A
2
+
B
2
+
C
2
)
−
1
2
,
m
=
(
a
2
A
2
+
b
2
B
2
+
c
2
C
2
)
1
2
.
h = (A^2 + B^2 + C^2)^{\frac{-1}{2}}, m = (a^2A^2 + b^2B^2 + c^2C^2)^{\frac{1}{2}}.
h
=
(
A
2
+
B
2
+
C
2
)
2
−
1
,
m
=
(
a
2
A
2
+
b
2
B
2
+
c
2
C
2
)
2
1
.
What is the algebraic condition for the plane not to intersect the ellipsoid?
Putnam