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  4. The locus of the midpoints of all chords of the parabola y2 = 4ax through its vertex is another parabola with directrix

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Q. The locus of the midpoints of all chords of the parabola $y^2 = 4ax$ through its vertex is another parabola with directrix

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Solution:

$2h=at^{2}, 2k=2at \Rightarrow t-k/a, \Rightarrow 2h =a \frac{k^{2}}{a^{2}}, y^{2}=2ax$, Equation of its directr $x=-a/2$

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