Question

If a2 +b2+c2-ab-bc-ca=o , Prove that a=b=c

Answer 1

Hey there !!!!!

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a²+b²+c²-ab-bc-ca=0

a²+b²+c²=ab+bc+ca---Equation 1

Multiplying equation 1 with 2 on both sides


2(a²+b²+c²)=2(ab+bc+ca)

2a²+2b²+2c²=2ab+2bc+2ca

a²+a²+b²+b²+c²+c²=2ab+2bc+2ca

Arranging like terms in form of X²-2XY+Y²=(X-Y)²

(a²-2ab+b²)+(b²-2bc+c²)+(c²-2ac+a²)=0

(a-b)²+(b-c)²+(c-a)²=0----Equation 2

Now square of any number is always positive.But according to equation 2 sum of squares is zero.This is possible only when each term is "0".

So,

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

⇒⇒a=b       ⇒⇒b=c        ⇒⇒c=a

From above a=b=c

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Hope this helped you.............