Find the directrix of the locus of the midpoi | vedclass

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(C) Let the variable point on the parabola $y^{2} = 4ax$ be $P(at^{2}, 2at)$ and the focus be $S(a, 0)$.Let $M(h, k)$ be the midpoint of $PS$.Then $h

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$x = 0$

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(C) Let the variable point on the parabola $y^{2} = 4ax$ be $P(at^{2}, 2at)$ and the focus be $S(a, 0)$.Let $M(h, k)$ be the midpoint of $PS$.Then $h = \frac{at^{2} + a}{2}$ and $k = \frac{2at + 0}{2} = at$.From $k = at$,we get $t = \frac{k}{a}$.Substituting $t$ into the equation for $h$: $h = \frac{a(k/a)^{2} + a}{2} = \frac{k^{2}/a + a}{2} = \frac{k^{2} + a^{2}}{2a}$.Thus,$2ah = k^{2} + a^{2}$,which simplifies to $k^{2} = 2a(h - a/2)$.The locus of $(h, k)$ is $y^{2} = 2a(x - a/2)$.Comparing this with the standard form $Y^{2} = 4AX$,where $Y = y$,$X = x - a/2$,and $4A = 2a \Rightarrow A = a/2$.The directrix is given by $X = -A$,so $x - a/2 = -a/2$.Therefore,$x = 0$.

Find the directrix of the locus of the midpoint of the line segment joining the focus and a variable point on the parabola $y^{2} = 4ax$.

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