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Mathematics
Mathematics
Problem
Prove that the locus of the middle points of normal chords of the ellipse
x
2
a
2
+
y
2
b
2
=
1
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
a
2
x
2
+
b
2
y
2
=
1
is
the curve
(
x
2
a
2
+
y
2
b
2
)
2
(
a
6
x
2
+
b
6
y
2
)
=
(
a
2
−
b
2
)
2
\text{the curve } \left( \frac{x^2}{a^2} + \frac{y^2}{b^2} \right)^2 \left( \frac{a^6}{x^2} + \frac{b^6}{y^2} \right) = (a^2 - b^2)^2
the curve
(
a
2
x
2
+
b
2
y
2
)
2
(
x
2
a
6
+
y
2
b
6
)
=
(
a
2
−
b
2
)
2
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