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CBSE Class 12-science Answered

find the equation of the plane passing through points A (a,0,0) , B (0,b,0) and C (0,0,c) . Reduce it to normal form. If plane ABC is at a distance of p from origion prove that (1/p^2) = (1/a^2) + (1/b^2) + (1/c^2)
Asked by srivastava.amrendra | 15 Oct, 2018, 11:50: PM
Expert Answer
space L e t space P comma space Q space a n d space R space b e space t h e space t h r e e space p o i n t s space w i t h space c o o r d i n a t e s space left parenthesis x 1 comma space y 1 comma space z 1 right parenthesis comma space left parenthesis x 2 comma space y 2 comma space z 2 right parenthesis comma space left parenthesis x 3 comma space y 3 comma space z 3 right parenthesis space r e s p e c t i v e l y. space T h e n space t h e space e q u a t i o n space o f space p l a n e space i s space a space asterisk times space left parenthesis x space – space x 0 right parenthesis space plus space b space asterisk times space left parenthesis y space – space y 0 right parenthesis space plus space c space asterisk times space left parenthesis z space – space z 0 right parenthesis space equals space 0 comma space w h e r e space a comma space b comma space c space a r e space d i r e c t i o n space r a t i o s space o f space n o r m a l space t o space t h e space p l a n e space a n d space left parenthesis x 0 comma space y 0 comma space z 0 right parenthesis space a r e space c o minus o r d i n a t e s space o f space a n y space p o i n t left parenthesis i. e space P comma space Q comma space o r space R right parenthesis space p a s sin g space t h r o u g h space t h e space p l a n e.
t o space g i n d space d i r c e t i o n space r a t i o space o f space n o r m a l space l e t s space f i n d space n o r m a l space v e c t o r space t o space top enclose A B space end enclose space a n d space space top enclose B C. end enclose space space N o r m a l space v e c t o r space t o space A B space a n d space B C space i s space top enclose A B space end enclose space X space bold space top enclose bold B bold C bold. end enclose
bold space bold space top enclose bold B bold C bold space end enclose bold space bold equals bold space bold italic C bold minus bold italic B bold equals bold left parenthesis bold 0 bold comma bold 0 bold comma bold italic c bold right parenthesis bold minus bold left parenthesis bold 0 bold comma bold italic b bold comma bold 0 bold right parenthesis bold equals bold left parenthesis bold 0 bold comma bold minus bold italic b bold comma bold italic c bold right parenthesis
bold space top enclose bold A bold B bold space end enclose bold equals bold space bold italic B bold minus bold italic C bold equals bold space bold left parenthesis bold 0 bold comma bold italic b bold comma bold 0 bold right parenthesis bold minus bold left parenthesis bold italic a bold comma bold 0 bold comma bold 0 bold right parenthesis bold equals bold left parenthesis bold minus bold italic a bold comma bold italic b bold comma bold 0 bold right parenthesis
bold space top enclose bold A bold B bold space end enclose bold space bold italic X bold space bold space top enclose bold BC bold equals bold left square bracket table row bold i bold j bold k row cell bold minus bold a end cell bold b bold 0 row bold 0 cell bold minus bold b end cell bold c end table bold right square bracket bold space bold equals bold italic i bold left parenthesis bold italic b bold italic c bold right parenthesis bold minus bold italic j bold left parenthesis bold minus bold italic a bold italic c bold right parenthesis bold plus bold italic k bold left parenthesis bold italic a bold italic b bold right parenthesis bold equals bold italic b bold italic c stack bold i bold space with bold overbrace on top bold space bold plus bold italic a bold italic c bold j with bold overbrace on top bold space bold plus bold space bold italic a bold italic b bold space bold k with bold overbrace on top bold space
bold italic h bold italic e bold italic n bold italic c bold italic e bold space bold italic d bold italic i bold italic r bold italic e bold italic c bold italic t bold italic i bold italic o bold italic n bold space bold italic r bold italic a bold italic t bold italic i bold italic o bold space bold italic o bold italic f bold space bold italic n bold italic o bold italic r bold italic m bold italic a bold italic l bold space bold italic i bold italic s bold space bold left parenthesis bold italic b bold italic c bold comma bold italic a bold italic c bold comma bold italic a bold italic b bold right parenthesis
bold italic e bold italic q bold italic u bold italic a bold italic t bold italic i bold italic o bold italic n bold space bold italic o bold italic f bold space bold italic p bold italic l bold italic a bold italic n bold italic e bold space bold italic i bold italic s bold space bold italic b bold italic c bold left parenthesis bold italic x bold minus bold italic a bold right parenthesis bold plus bold space bold italic a bold italic c bold space bold left parenthesis bold italic y bold minus bold 0 bold right parenthesis bold plus bold italic a bold italic b bold left parenthesis bold italic z bold minus bold 0 bold right parenthesis bold equals bold 0
bold italic b bold italic c bold italic x bold plus bold italic a bold italic c bold italic y bold plus bold italic a bold italic b bold italic z bold equals bold italic a bold italic b bold italic c bold minus bold minus bold minus bold minus bold minus bold minus bold minus bold minus bold left parenthesis bold 1 bold right parenthesis

bold italic n bold italic o bold italic r bold italic m bold italic a bold italic l bold space bold italic f bold italic o bold italic r bold italic m bold space bold italic i bold italic s bold space stack bold r bold. with bold bar on top bold n with bold hat on top bold equals bold italic p bold space bold italic w bold italic h bold italic e bold italic r bold space bold italic p bold space bold italic i bold italic s bold space bold italic t bold italic h bold italic e bold space bold italic d bold italic i bold italic s bold tan bold italic c bold italic e bold space bold italic f bold italic r bold italic o bold italic m bold space bold italic t bold italic h bold italic e bold space bold italic o bold italic r bold italic i bold italic g bold italic i bold italic n bold.
bold italic n bold italic o bold italic r bold italic m bold italic a bold italic l bold space bold italic v bold italic e bold italic c bold italic t bold italic o bold italic r bold space bold n with bold rightwards harpoon with barb upwards on top bold space bold italic i bold italic s bold space bold left parenthesis bold italic b bold italic c bold comma bold italic a bold italic c bold comma bold italic a bold italic b bold right parenthesis bold space bold italic s bold italic o bold space bold n with bold hat on top bold equals fraction numerator bold 1 over denominator square root of bold left parenthesis bold b bold c bold right parenthesis to the power of bold 2 bold plus bold left parenthesis bold a bold c bold right parenthesis to the power of bold 2 bold plus bold left parenthesis bold a bold b bold right parenthesis to the power of bold 2 end root end fraction bold space bold left parenthesis bold italic b bold italic c bold comma bold italic a bold italic c bold comma bold italic a bold italic b bold right parenthesis
stack bold r bold. with bold bar on top bold. fraction numerator bold 1 over denominator square root of bold left parenthesis bold bc bold right parenthesis to the power of bold 2 bold plus bold left parenthesis bold ac bold right parenthesis to the power of bold 2 bold plus bold left parenthesis bold ab bold right parenthesis to the power of bold 2 end root end fraction bold space bold left parenthesis bold italic b bold italic c bold i with bold hat on top bold plus bold italic a bold italic c bold j with bold hat on top bold plus bold italic a bold italic b bold k with bold hat on top bold right parenthesis bold equals bold italic p
bold left parenthesis bold italic x bold i with bold hat on top bold plus bold italic y bold j with bold hat on top bold plus bold italic z bold k with bold hat on top bold space bold right parenthesis bold. fraction numerator bold 1 over denominator square root of bold left parenthesis bold bc bold right parenthesis to the power of bold 2 bold plus bold left parenthesis bold ac bold right parenthesis to the power of bold 2 bold plus bold left parenthesis bold ab bold right parenthesis to the power of bold 2 end root end fraction bold space bold left parenthesis bold italic b bold italic c bold i with bold hat on top bold plus bold italic a bold italic c bold j with bold hat on top bold plus bold italic a bold italic b bold k with bold hat on top bold right parenthesis bold equals bold italic p
bold italic b bold italic c bold italic x bold plus bold italic a bold italic c bold italic y bold plus bold italic a bold italic b bold italic z bold equals bold italic p bold. square root of bold left parenthesis bold bc bold right parenthesis to the power of bold 2 bold plus bold left parenthesis bold ac bold right parenthesis to the power of bold 2 bold plus bold left parenthesis bold ab bold right parenthesis to the power of bold 2 end root

bold italic c bold italic o bold italic m bold italic p bold italic a bold italic r bold italic i bold italic n bold italic g bold space bold italic w bold italic i bold italic t bold italic h bold space bold italic a bold italic b bold italic o bold italic v bold italic e bold space bold left parenthesis bold 1 bold right parenthesis
bold italic w bold italic e bold space bold italic g bold italic e bold italic t bold space bold italic p bold. square root of bold left parenthesis bold bc bold right parenthesis to the power of bold 2 bold plus bold left parenthesis bold ac bold right parenthesis to the power of bold 2 bold plus bold left parenthesis bold ab bold right parenthesis to the power of bold 2 end root bold equals bold italic a bold italic b bold italic c
bold italic s bold italic q bold italic u bold italic a bold italic r bold italic i bold italic n bold italic g bold space bold italic o bold italic n bold space bold italic b bold italic o bold italic t bold italic h bold space bold italic s bold italic i bold italic d bold italic e bold space bold italic w bold italic e bold space bold italic g bold italic e bold italic t bold space bold italic p to the power of bold 2 bold. bold left parenthesis bold left parenthesis bold bc bold right parenthesis to the power of bold 2 bold plus bold left parenthesis bold ac bold right parenthesis to the power of bold 2 bold plus bold left parenthesis bold ab bold right parenthesis to the power of bold 2 bold right parenthesis bold equals bold a to the power of bold 2 bold b to the power of bold 2 bold c to the power of bold 2
bold divinding bold space bold with bold space bold a to the power of bold 2 bold b to the power of bold 2 bold c to the power of bold 2 bold space bold on bold space bold both bold space bold sides bold space bold we bold space bold get bold space bold p to the power of bold 2 bold. bold left parenthesis bold 1 over bold a to the power of bold 2 bold plus bold 1 over bold b to the power of bold 2 bold plus bold 1 over bold c to the power of bold 2 bold right parenthesis bold equals bold 1
bold so bold space bold left parenthesis bold 1 over bold a to the power of bold 2 bold plus bold 1 over bold b to the power of bold 2 bold plus bold 1 over bold c to the power of bold 2 bold right parenthesis bold equals bold 1 over bold p to the power of bold 2

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