Find the locus of the midpoints of all focal | vedclass

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(C) Let the midpoint of the focal chord be $(h, k)$.The equation of a chord of the parabola $y^2 = 4ax$ with midpoint $(h, k)$ is given by $T = S_1$,w

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$y^2 = 2a(x - a)$

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(C) Let the midpoint of the focal chord be $(h, k)$.The equation of a chord of the parabola $y^2 = 4ax$ with midpoint $(h, k)$ is given by $T = S_1$,where $T = yk - 2a(x + h)$ and $S_1 = k^2 - 4ah$.So,the equation is $yk - 2a(x + h) = k^2 - 4ah$.Since this chord passes through the focus $(a, 0)$,we substitute $x = a$ and $y = 0$ into the equation:$0(k) - 2a(a + h) = k^2 - 4ah$$-2a^2 - 2ah = k^2 - 4ah$$k^2 = 2ah - 2a^2$$k^2 = 2a(h - a)$.Replacing $(h, k)$ with $(x, y)$,the locus is $y^2 = 2a(x - a)$.

$A. \ 2x-5=0$	$I. \ \text{Vertex}$
$B. \ (\frac{3}{2}, -1)$	$II. \ \text{Focus}$
$C. \ y+1=0$	$III. \ \text{Equation of directrix}$
$D. \ (2, -1)$	$IV. \ \text{Equation of the axis}$
	$V. \ \text{Equation of the Latus rectum}$

Find the locus of the midpoints of all focal chords of the parabola $y^2 = 4ax$.

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