Question

if bc+ca+ab=0 ,then prove that 1/(a^2-bc)+1/(b^2-ca)+1/(c^2-ab)=0

Answer 1

Given ab+bc+ca=0
1/(a2-bc ) + 1/(c2-ab) + 1/(a2-bc)
Put -ac = ab + bc ; -ab = ac+ bc and -bc =ab + ac

=(1/(a2+ab+ac))+(1/(c2+ac+bc))+(1/(a2+ab+ac))
=(1/a(a+b+c))+(1/b(a+b+c))+(1/c(a+b+c))
By taking LCM we get
=(ab + bc + ca)/[(a+b+c)(abc)]
put, ab+bc+ca=0
= 0/[(a+b+c)(abc)] = 0