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Prove that the locus of the centre of th...

Prove that the locus of the centre of the circle, which passes through the vertex of the parabola `y^2 = 4ax` & through its intersection with a normal chord is `2y^2 = ax - a^2`.

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Show that the locus of the middle points of chords of the parabola y^(2) = 4ax passing through the vertex is the parabola y^(2) = 2ax

If (a ,b) is the midpoint of a chord passing through the vertex of the parabola y^2=4x, then prove that 2a=b^2