The locus of the midpoints of the chords of t | vedclass

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Question

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

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$y^2 = 2ax$

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

The locus of the midpoints of the chords of the parabola $y^2 = 4ax$ which pass through the origin is:

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