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Two systems of rectangular axis have the same origin. If plane cuts them at distances a, b, c and a’, b’, c, respectively from the origin, then, prove that 1/a2 + 1/b2  + 1/c2 = 1/a’2 + 1/b’2  + 1/c’2

Asked by sourabh7jain 10th January 2017, 6:37 PM
Answered by Expert
Answer:
begin mathsize 20px style We space know space that space the space perpendicular space distance space of space the space plane space ax space plus space by space plus space cz space plus space straight d space equals space 0 space from space the space origin space is space given space by space open vertical bar fraction numerator straight d over denominator square root of straight a squared plus straight b squared plus straight c squared end root end fraction close vertical bar.
Also comma space if space two space system space of space lines space have space the space same space origin space then space their space perpendicular space distance space from space the space origin space to space the space plane space in space
both space the space systems space are space equal.

Let space the space equation space of space the space plane space in space both space the space systems space be space
straight x over straight a plus straight y over straight b plus straight z over straight c equals 1 space space space and space space space space fraction numerator straight X over denominator straight a apostrophe end fraction plus fraction numerator straight Y over denominator straight b apostrophe end fraction plus fraction numerator straight Z over denominator straight c apostrophe end fraction equals 1 space
Recall space that space the space perpendicular space distance space of space the space plane space ax space plus space by space plus space cz space plus space straight d space equals space 0 space from space the space origin space is space space open vertical bar fraction numerator straight d over denominator square root of straight a squared plus straight b squared plus straight c squared end root end fraction close vertical bar..
It space is space given space that space the space origin space is space same space for space both space the space systems.
Hence space the space perpendicular space distance space of space the space plane space from space the space origin space are space equal space for space both space the space systems.
That space is comma space open vertical bar fraction numerator negative 1 over denominator square root of begin display style 1 over straight a squared plus 1 over straight b squared plus 1 over straight c squared end style end root end fraction close vertical bar equals open vertical bar fraction numerator negative 1 over denominator square root of begin display style fraction numerator 1 over denominator straight a apostrophe squared end fraction plus fraction numerator 1 over denominator straight b apostrophe squared end fraction plus fraction numerator 1 over denominator straight c apostrophe squared end fraction end style end root end fraction close vertical bar
rightwards double arrow fraction numerator 1 over denominator square root of begin display style 1 over straight a squared plus 1 over straight b squared plus 1 over straight c squared end style end root end fraction equals fraction numerator 1 over denominator square root of begin display style fraction numerator 1 over denominator straight a apostrophe squared end fraction plus fraction numerator 1 over denominator straight b apostrophe squared end fraction plus fraction numerator 1 over denominator straight c apostrophe squared end fraction end style end root end fraction
rightwards double arrow square root of 1 over straight a squared plus 1 over straight b squared plus 1 over straight c squared end root equals square root of fraction numerator 1 over denominator straight a apostrophe squared end fraction plus fraction numerator 1 over denominator straight b apostrophe squared end fraction plus fraction numerator 1 over denominator straight c apostrophe squared end fraction end root
rightwards double arrow 1 over straight a squared plus 1 over straight b squared plus 1 over straight c squared equals fraction numerator 1 over denominator straight a apostrophe squared end fraction plus fraction numerator 1 over denominator straight b apostrophe squared end fraction plus fraction numerator 1 over denominator straight c apostrophe squared end fraction space space space.... space left parenthesis Squaring space both space sides right parenthesis
Hence space proved. end style
Answered by Expert 11th January 2017, 9:34 AM
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