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While studying conic sections, in the parabola portion, I read that

The sum of the ordinates of the extremities of the chords of the parabola $y^2=4.a.x$ which are parallel to each other is constant.

But I was in doubt, how is this possible, and what is this $constant$ $value$? I would be obliged if anyone can help me with this. Thanks

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Consider a set of parallel chords having slope $m$. The equation of all these chords is of the form $y=mx+c$

The ordinates of the extremities of a chord will satisfy the following equation: $$y^2=4a\left({y-c\over m}\right)\implies my^2-4ay+4ac=0$$

The sum of the ordinates is just the sum of roots of that quadratic equation, which is $\dfrac{4a}{m}$

Also note that this constancy is precisely the reason that the diameters of any parabola are always parallel to its axis.

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  • $\begingroup$ So it is true for any number of chords I draw parallel to each other? $\endgroup$ – Aneek Dec 15 '15 at 13:59
  • $\begingroup$ @Aneek yes.${}{}{}$ $\endgroup$ – G-man Dec 15 '15 at 14:00
  • $\begingroup$ Well, you are brilliant, thanks! $\endgroup$ – Aneek Dec 15 '15 at 14:00

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