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

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(C) Let the focus of the parabola $y^{2}=4ax$ be $S(a, 0)$ and a moving point on the parabola be $P(at^{2}, 2at)$.Let $M(h, k)$ be the mid-point of th

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$x=0$

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(C) Let the focus of the parabola $y^{2}=4ax$ be $S(a, 0)$ and a moving point on the parabola be $P(at^{2}, 2at)$.Let $M(h, k)$ be the mid-point of the segment $SP$.Then,$h = \frac{at^{2}+a}{2}$ and $k = \frac{2at+0}{2} = at$.From $k = at$,we have $t = \frac{k}{a}$.Substituting this into the expression for $h$:$h = \frac{a(\frac{k}{a})^{2}+a}{2} = \frac{\frac{k^{2}}{a}+a}{2} = \frac{k^{2}+a^{2}}{2a}$.$2ah = k^{2}+a^{2} \Rightarrow k^{2} = 2ah - a^{2} = 2a(h - \frac{a}{2})$.Replacing $(h, k)$ with $(x, y)$,the locus is $y^{2} = 2a(x - \frac{a}{2})$.This is a parabola of the form $Y^{2} = 4AX$,where $Y=y$,$X=x-\frac{a}{2}$,and $4A = 2a \Rightarrow A = \frac{a}{2}$.The directrix of $Y^{2} = 4AX$ is $X = -A$.Substituting the values: $x - \frac{a}{2} = -\frac{a}{2} \Rightarrow x = 0$.

The locus of the mid-point of the line segment joining the focus of the parabola $y^{2}=4ax$ to a moving point of the parabola,is another parabola whose directrix is

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