Question

If a^2 + b^2 + c^2 - ab- bc - ca=0, prove that a = b= c

Answer 1

Answer is given below:

a^2 + b^2 + c^2 - ab- bc - ca=0

Multiply Both Sides with 2,We get,

2(a^2 + b^2 + c^2 - ab- bc - ca)=0

2a^2 + 2b^2 + 2c^2 - 2ab- 2bc - 2ca=0

(a^2 -2ba+ b^2 )+( b^2 -2bc+ c^2 )+(c^2 -2a+ a^2)=0

(a-b)^2+(b-c)^2+(c-a)^2=0

Since, the Sum of the square is 0.Then each term should be 0,

(a-b)^2=0,(b-c)^2=0,(c-a)^2=0

a-b=0,b-c=0,c-a=0                   ...................(1)

Hence,From equation (1), we get,

a=b&b=c&c=a                           ...................(2)  

Hence,From equation (2), we get,

a=b=c

Hence, proved

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