if a2/(b+c) +b2/(c+a) +c2/(a+b)=0 then prove that a/(b+c) +b/(c+a) +c/(a+b) =1
khokan , 8 Years ago
Grade 12th pass
Algebra> if a2/(b+c) +b2/(c+a) +c2/(a+b)=0 then pr...
2 Answers
Arun
Dear student
take a/(b+c) +b/(c+a) +c/(a+b) =1
then multiply (a+b+c) in both the sides
a(a+b+c) / (b+c) + b(a+b+c) / (a+c) + c(a+b+c) / (b+a) = a+b+c
a2 / (b+c) + a(b+c)/ (b+c) + b2 / (c+a) + b(c+a)/ (c+a) + c2 / (a+b) + c(a+b)/ (a+b) = a+b+c
a2 / (b+c) + a + b2 / (c+a) + b + c2 / (a+b) + c = a+b+c
a2 / (b+c) + b2 / (c+a) + c2 / (a+b) = 0
Hence Proved
Regards
Arun (askIITians forum expert)
Sayantan Garai
a²/b+c + b²/c+a + c²/a+b = 0
=> a²/b+c + a + b²/c+a + b + c²/a+b + c = a+b+c
=> a²+ab+ac/b+c + b²+bc+ab/c+a + c²+ac+bc/a+b = a+b+c
=> (a+b+c)(a/b+c + b/c+a + c/a+b ) = a+b+c
=> a/b+c + b/c+a + c/a+b = 1
=> a²/b+c + a + b²/c+a + b + c²/a+b + c = a+b+c
=> a²+ab+ac/b+c + b²+bc+ab/c+a + c²+ac+bc/a+b = a+b+c
=> (a+b+c)(a/b+c + b/c+a + c/a+b ) = a+b+c
=> a/b+c + b/c+a + c/a+b = 1
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