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Grade: 10
        
a+b2+c2=ab+bc+ca then prove that 
a/b+c +b/c+a +c/b+a =3/2
6 months ago

Answers : (2)

Vikas TU
11678 Points
							
Dear student 
a2 + b2 + c2 – ab – bc – ca = 0
Multiply both sides with 2, we get
2( a2 + b2 + c2 – ab – bc – ca) = 0
⇒ 2a2 + 2b2 + 2c2 – 2ab – 2bc – 2ca = 0
⇒ a2 + a2 + b2 + b2 + c2 + c2– 2ab – 2bc – 2ca = 0
⇒ (a2 – 2ab + b2) + (b2 – 2bc + c2) + (c2 – 2ca + a2) = 0
⇒ (a – b)2 + (b – c)2 + (c – a)2 = 0
Since the sum of squares is zero, it means each term should be zero.
⇒ (a – b)2 = 0,  (b – c)2 = 0, (c – a)2 = 0
⇒ a – b = 0,  b – c = 0,  c – a = 0
⇒ a = b,  b = c, c = a
⇒ a = b = c
Put this value in the equation , 
a/b+c +b/c+a +c/b+a =3/2
Hope this will help 
Good Luck 
Cheers 
6 months ago
Arun
24735 Points
							
 
take it as
a2+b2+c2-ab-bc-ca=0
Multiply both sides with 2, we get
2( a2 + b2 + c2 – ab – bc – ca) = 0
2a2 + 2b2 + 2c2 – 2ab – 2bc – 2ca = 0
(a2 – 2ab + b2) + (b2 – 2bc + c2) + (c– 2ca + a2) = 0
(a –b)2 + (b – c)2 + (c – a)2 = 0
Since the sum of square is zero then each term should be zero
(a –b)2 = 0,  (b – c)2 = 0, (c – a)2 = 0
(a –b) = 0,  (b – c) = 0, (c – a) = 0
a = b,  b = c, c = a
a = b = c.
therefore c+a/b=2
 
similarly all the terms = ½
hence answer = 1/2+ ½ + ½ = 3/2
6 months ago
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