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If-a-b-2-b-c-2-c-a-2-0-then-the-values-of-a-b-and-c-are-equal-

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Question Number 5018 by lec123 last updated on 02/Apr/16
If (a−b)^2 +(b−c)^2 +(c−a)^2 =0, then the values of a, b and c are equal.
If(a−b)2+(b−c)2+(c−a)2=0,thenthevaluesofa,bandcareequal.
Commented by Rasheed Soomro last updated on 03/Apr/16
a=b=c only when a,b,c ∈ R.
a=b=conlywhena,b,c∈R.
Answered by prakash jain last updated on 02/Apr/16
If (a−b)^2 is a whole square so (a−b)^2 ≥0. Similarly (b−c)^2 ≥0 and (c−a)^2 ≥0 Given (a−b)^2 +(b−c)^2 +(c−a)^2 ≥0 Each of the terms have to be equal to 0. a−b=0⇒a=b b−c=0⇒b=c c−a=0⇒c=a Hence a=b=c
If(a−b)2isawholesquareso(a−b)2⩾0.Similarly(b−c)2⩾0and(c−a)2⩾0Given(a−b)2+(b−c)2+(c−a)2⩾0Eachofthetermshavetobeequalto0.a−b=0⇒a=bb−c=0⇒b=cc−a=0⇒c=aHencea=b=c

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