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If a+b+c=0 then prove that1/(2a ^ +bc) +1/(2b^2+ca)+1/(2c^2+ab)=0

If a+b+c=0 then prove that1/(2a ^ +bc) +1/(2b^2+ca)+1/(2c^2+ab)=0

Grade:10

2 Answers

Arun
25750 Points
6 years ago
Dear studentI think you have tyoed the question wrong.Please repost it.RegardsArun (AskIITians forum expert)
prakash ojha
17 Points
4 years ago
2a²+bc=2(b+c)²+bc
             =2(b²+2bc+c²)+bc
             =2b²+4bc+2c²+bc
             =2b²+5bc+2c²
             =(2b+c)(b+2c)
             =(b+b+c)(b+c+c)
             =(b-a)(-a+c) [since a+b+c=0]
             =(b-a)(c-a)
Similarly,
2b²+ca=(c-b)(a-b)
and
2c²+ab=(a-c)(b-c)
Now,
1/2a²+bc + 1/2b²+ca + 1/2c²+ab
=1/(b-a)(c-a) + 1/(c-b)(a-b) + 1/(a-c)(b-c)
=1/(a-b)(a-c) -1/(b-c)(a-b) + 1/(a-c)(b-c)
={b-c-(a-c)+a-b)}/(a-b)(a-c)(b-c)
={b-c-a+c+a-b}/(a-b)(
 
 
 
 

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