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Mathematics
Problem
Prove that the limiting points of the system
x
2
+
y
2
+
2
g
x
+
c
+
λ
(
x
2
+
y
2
+
2
f
x
+
k
)
=
0
x^2 + y^2 + 2gx + c + \lambda(x^2 + y^2 + 2fx + k) = 0
x
2
+
y
2
+
2
g
x
+
c
+
λ
(
x
2
+
y
2
+
2
f
x
+
k
)
=
0
subtend a right angle at the origin, if
c
g
2
+
k
f
2
=
2
\frac{c}{g^2} + \frac{k}{f^2} = 2
g
2
c
+
f
2
k
=
2
.
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