Question

The value of x in $\frac{x-bc}{b+c}+\frac{x-ca}{c+a}+\frac{x-ab}{a+b}=a+b+c$ is

• A. $a+b+c$
• B. abc
• C. $ab+bc+ca$
• D. $a^2+b^2+c^2$

Answer 1

The correct option is C $ab+bc+ca$

The solution in this problem can be given by back-substitution of options.
1) For option A:
LHS = $\frac{a+b+c-bc}{b+c}+\frac{a+b+c-ca}{c+a}+\frac{a+b+c-ab}{a+b}$
which gives atleast one product term. So A cannot be the correct option.
2) For option B:
LHS = $\frac{abc-bc}{b+c}+\frac{abc-ca}{c+a}+\frac{abc-ab}{a+b}$
which also gives product terms. So B is also not correct.
3) For option c:
LHS = $\frac{ab+bc+ca-bc}{b+c}+\frac{ab+bc+ca-ca}{c+a}+\frac{ab+bc+ca-ab}{a+b}$
= $\frac{a(b+c)}{b+c}+\frac{b(a+c)}{c+a}+\frac{c(a+b)}{a+b}$
= $a+b+c$, which is the required term in RHS.
Hence, the correct option is C.