Question : If a + b + c = 0, then the value of (a + b – c)2 + ( b + c – a)2 + ( c + a – b)2 is:
Option 1: 0
Option 2: 8abc
Option 3: 4(a2 + b2 + c2)
Option 4: 4(ab + bc + ca)
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Correct Answer: 4(a 2 + b 2 + c 2 )
Solution :
a + b + c=0
⇒ a + b = –c --------------(i)
⇒ b + c = –a --------------(ii)
⇒ c + a = –b --------------(iii)
Substituting (i), (ii) and (iii) in (a + b – c)
2
+ (b + c – a)
2
+ (c + a – b)
2
, we get
(a + b – c)
2
+ (b + c – a)
2
+ (c + a – b)
2
= (–c –c)
2
+ (–a –a)
2
+ ( –b –b)
2
= 4c
2
+ 4a
2
+ 4b
2
= 4(a
2
+ b
2
+ c
2
)
Hence, the correct answer is 4(a
2
+ b
2
+ c
2
).
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