Prove-that-the-two-parabola-y-2-4ax-and-y-2-4c-x-b-cannot-have-a-common-normal-other-than-the-axis-unless-b-a-c-gt-2-

Tinku Tara June 4, 2023 Coordinate Geometry 0 Comments

Question Number 51901 by peter frank last updated on 31/Dec/18

Prove that the two parabola y^2 =4ax and y^2 =4c(x−b) cannot have a common normal other than the axis unless (b/(a−c))>2

$${Prove}\:{that}\:{the}\:{two}\: \\ $$$${parabola}\:{y}^{\mathrm{2}} =\mathrm{4}{ax}\:{and} \\ $$$${y}^{\mathrm{2}} =\mathrm{4}{c}\left({x}−{b}\right)\:{cannot} \\ $$$${have}\:{a}\:{common}\:{normal} \\ $$$${other}\:{than}\:{the}\:{axis}\:{unless} \\ $$$$\frac{{b}}{{a}−{c}}>\mathrm{2} \\ $$

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Prove-that-the-two-parabola-y-2-4ax-and-y-2-4c-x-b-cannot-have-a-common-normal-other-than-the-axis-unless-b-a-c-gt-2-

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