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Question Number 11061 by FilupS last updated on 10/Mar/17

A 3 dimentional parabola f(x, y, z) has a focus P(x, y, z)=(p, q, r). A rectangular box with side lengths a, b, and c (see diagram), has a point P on the front center point of its face. If the origin lies on the opposite face, in the middle, what is the function of the curve? Note Origin O(x, y, z)=(0, 0, 0) O(x, y, z) = (p−c, q, r)

$$\mathrm{A}\:\mathrm{3}\:\mathrm{dimentional}\:\mathrm{parabola}\:{f}\left({x},\:{y},\:{z}\right) \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{focus}\:{P}\left({x},\:{y},\:{z}\right)=\left({p},\:{q},\:{r}\right). \\ $$$$\: \\ $$$$\mathrm{A}\:\mathrm{rectangular}\:\mathrm{box}\:\mathrm{with}\:\mathrm{side}\:\mathrm{lengths} \\ $$$${a},\:{b},\:\mathrm{and}\:{c}\:\left(\mathrm{see}\:\mathrm{diagram}\right),\:\mathrm{has}\:\mathrm{a}\:\mathrm{point}\:{P} \\ $$$$\mathrm{on}\:\mathrm{the}\:\mathrm{front}\:\mathrm{center}\:\mathrm{point}\:\mathrm{of}\:\mathrm{its}\:\mathrm{face}. \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{origin}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{the}\:\mathrm{opposite}\:\mathrm{face}, \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{middle},\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{function}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{curve}? \\ $$$$\: \\ $$$$\boldsymbol{\mathrm{Note}} \\ $$$$\mathrm{Origin}\:{O}\left({x},\:{y},\:{z}\right)=\left(\mathrm{0},\:\mathrm{0},\:\mathrm{0}\right) \\ $$$${O}\left({x},\:{y},\:{z}\right)\:=\:\left({p}−{c},\:{q},\:{r}\right) \\ $$

Commented by FilupS last updated on 10/Mar/17

Commented by FilupS last updated on 10/Mar/17

Let c be parallel to the x axis Let a be parallel to the y axis Let b be parallel to the z axis

$$\mathrm{Let}\:{c}\:\mathrm{be}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:{x}\:\mathrm{axis} \\ $$$$\mathrm{Let}\:{a}\:\mathrm{be}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:{y}\:\mathrm{axis} \\ $$$$\mathrm{Let}\:{b}\:\mathrm{be}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the}\:{z}\:\mathrm{axis} \\ $$

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