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Putnam
1940 Putnam
B4
B4
Part of
1940 Putnam
Problems
(1)
Putnam 1940 B4
Source: Putnam 1940
2/22/2022
Prove that the locus of the point of intersection of three mutually perpendicular planes tangent to the surface
a
x
2
+
b
y
2
+
c
z
2
=
1
(
where
a
b
c
≠
0
)
ax^2 + by^2 +cz^2 =1\;\;\; (\text{where}\;\;abc \ne 0)
a
x
2
+
b
y
2
+
c
z
2
=
1
(
where
ab
c
=
0
)
is the sphere
x
2
+
y
2
+
z
2
=
1
a
+
1
b
+
1
c
.
x^2 +y^2 +z^2 =\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.
x
2
+
y
2
+
z
2
=
a
1
+
b
1
+
c
1
.
Putnam
geometry
3D geometry
sphere