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A set of parallel chords of the parabola $y^2=4 a x$ have their mid-point on
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Any straight line parallel to the axis
Let $y=m x+c$ is chord and $c$ is variable
$\Rightarrow x=\left(\frac{y-c}{m}\right)$ by $y^2=4 a x$
For getting points of intersection,
$y^2=4 a\left(\frac{y-c}{m}\right) \Rightarrow$ $y^2-\frac{4 a y}{m}+\frac{4 a c}{m}=0$
$\Rightarrow y_1+y_2=\frac{4 a}{m} \Rightarrow \frac{y_1+y_2}{2}=\frac{2 a}{m}$
which is a constant; independent to $c$.
$\Rightarrow x=\left(\frac{y-c}{m}\right)$ by $y^2=4 a x$
For getting points of intersection,
$y^2=4 a\left(\frac{y-c}{m}\right) \Rightarrow$ $y^2-\frac{4 a y}{m}+\frac{4 a c}{m}=0$
$\Rightarrow y_1+y_2=\frac{4 a}{m} \Rightarrow \frac{y_1+y_2}{2}=\frac{2 a}{m}$
which is a constant; independent to $c$.
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