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Question

If a,b,c are real and a3+b3+c3=3abc and a+b+c0, then the relation between a,b,c will be:

A
a+b=c
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B
a+c=b
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C
a=b=c
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D
b+c=a
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Solution

The correct option is C a=b=c
if a,b,c are real, a3+b3+c3=3abc
given a+b+c0
Relation b/w a,b,c=?
a3+b3+c3=3abc
a3+b3+c33abc=0
fartrise the given expression
(a+b+c)(a2+b2+c2abbcca)=0
Given, a+b+c0
So, a2+b2+c2abbcca=0 (Must be, AS the product is Δ)
Therefore,
(a2+b2+c2abbcca)=0
Multiply by 2 on both the sides
(2a2+2b2+2c22ab2bc2ca)=0
Seprate individual square
a2+b22ab+b2+c22bc+c2+a22ca=0
(ab)2+(bc)2+(ca)2=0
(ab)2+(bc)2+(ca)2=0
We know (ab)2,(bc)2,(ca)2>0
So, for Add equal to 0, individual becomes 0
As ab=0
a=b...(1)
As bc=0
b=c...(2)
from equation (1) & (2)
a=b=c

1180504_803254_ans_273fc35c2d594428ae50547bd7fc2cf6.jpg

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