A set of parallel chords of the parabola y^2 | vedclass

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(C) Let the equation of the parallel chords be $y = mx + c$,where $m$ is constant and $c$ is a variable parameter.Substituting $x = \frac{y - c}{m}$ i

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Any straight line parallel to the axis

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(C) Let the equation of the parallel chords be $y = mx + c$,where $m$ is constant and $c$ is a variable parameter.Substituting $x = \frac{y - c}{m}$ into the parabola equation $y^2 = 4ax$,we get:$y^2 = 4a \left( \frac{y - c}{m} \right)$$my^2 - 4ay + 4ac = 0$$y^2 - \frac{4a}{m}y + \frac{4ac}{m} = 0$Let the points of intersection be $(x_1, y_1)$ and $(x_2, y_2)$. The sum of the roots is $y_1 + y_2 = \frac{4a}{m}$.The $y$-coordinate of the mid-point is $Y = \frac{y_1 + y_2}{2} = \frac{2a}{m}$.Since $Y = \frac{2a}{m}$ is a constant,the mid-points lie on a line parallel to the axis of the parabola $(y = 0)$.

$A$ set of parallel chords of the parabola $y^2 = 4ax$ have their mid-points on:

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