The locus of the mid-point of the line segmen | vedclass

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(C) Let the focus be $S(a, 0)$.Let any point on the parabola $y^{2}=4ax$ be $P(at^{2}, 2at)$.Let the mid-point of $SP$ be $(x, y)$.Then,$x = \frac{a +

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$x=0$

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(C) Let the focus be $S(a, 0)$.Let any point on the parabola $y^{2}=4ax$ be $P(at^{2}, 2at)$.Let the mid-point of $SP$ be $(x, y)$.Then,$x = \frac{a + at^{2}}{2}$ and $y = \frac{0 + 2at}{2} = at$.From $y = at$,we get $t = \frac{y}{a}$.Substituting $t$ into the equation for $x$:$x = \frac{a + a(\frac{y}{a})^{2}}{2} = \frac{a + \frac{y^{2}}{a}}{2}$.$2x = a + \frac{y^{2}}{a} \implies \frac{y^{2}}{a} = 2x - a \implies y^{2} = 2a(x - \frac{a}{2})$.This is a parabola of the form $Y^{2} = 4AX$ where $A = \frac{a}{2}$ and $X = x - \frac{a}{2}$.The directrix of $Y^{2} = 4AX$ is $X = -A$.Therefore,$x - \frac{a}{2} = -\frac{a}{2}$.$x = 0$.

The locus of the mid-point of the line segment joining the focus and any point on the parabola $y^{2}=4ax$ is a parabola. Find the equation of its directrix.

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