Find the locus of the midpoint of the chord o | vedclass

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(B) Let the vertex of the parabola $y^2 = 4ax$ be $O(0, 0)$.Let $Q(at^2, 2at)$ be a point on the parabola.The midpoint $P(x, y)$ of the chord $OQ$ is

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$y^2 = 2x$

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(B) Let the vertex of the parabola $y^2 = 4ax$ be $O(0, 0)$.Let $Q(at^2, 2at)$ be a point on the parabola.The midpoint $P(x, y)$ of the chord $OQ$ is given by:$x = \frac{0 + at^2}{2} = \frac{at^2}{2}$$y = \frac{0 + 2at}{2} = at$From the second equation,$t = \frac{y}{a}$.Substituting $t$ into the first equation:$x = \frac{a}{2} \left(\frac{y}{a}\right)^2 = \frac{a}{2} \cdot \frac{y^2}{a^2} = \frac{y^2}{2a}$$y^2 = 2ax$For the given parabola $y^2 = 4x$,we have $a = 1$.Substituting $a = 1$ into the locus equation $y^2 = 2ax$,we get $y^2 = 2x$.

Find the locus of the midpoint of the chord of the parabola $y^2 = 4x$ drawn from the vertex.

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