Prove that the locus of the middle points of normal chords of the parabola \( y^{2}=4 a x \) is ...

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Prove that the locus of the middle points of normal chords of the parabola \( y^{2}=4 a x \) is \( \frac{y^{2}}{2 a}+\frac{4 a^{2}}{y^{2}}=x-2 a \).
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