The locus of the mid-points of all chords of | vedclass

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(D) Let the chord pass through the vertex $V(0, 0)$ and intersect the parabola $y^{2}=4ax$ at $P(at^{2}, 2at)$.Let $(h, k)$ be the mid-point of the ch

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$x=-\frac{a}{2}$

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(D) Let the chord pass through the vertex $V(0, 0)$ and intersect the parabola $y^{2}=4ax$ at $P(at^{2}, 2at)$.Let $(h, k)$ be the mid-point of the chord $VP$.Then, $h = \frac{at^{2}+0}{2} = \frac{at^{2}}{2}$ and $k = \frac{2at+0}{2} = at$.From $k = at$, we have $t = \frac{k}{a}$.Substituting $t$ in the expression for $h$: $h = \frac{a}{2} \left(\frac{k}{a}\right)^{2} = \frac{k^{2}}{2a}$.Thus, $k^{2} = 2ah$.The locus of the mid-point $(h, k)$ is $y^{2} = 2ax$.This can be rewritten as $(y-0)^{2} = 4\left(\frac{a}{2}\right)(x-0)$.For a parabola $Y^{2} = 4AX$, the directrix is $X = -A$.Here, $A = \frac{a}{2}$, so the directrix is $x = -\frac{a}{2}$.

The locus of the mid-points of all chords of the parabola $y^{2}=4ax$ passing through its vertex is another parabola with directrix:

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