Question

If $x,y,z$ are positive real numbers and $a,b,c$ are rational numbers, then the value of $\frac{1}{1+x^{b-a}+x^{c-a}}+\frac{1}{1+x^{a-b}+x^{c-b}}+\frac{1}{1+x^{b-c}+x^{a-c}}$ is

• A. $-1$
• B. $0$
• C. $1$
• D. None of these

Answer 1

The correct option is C $1$

$\frac{1}{1+x^{b-a}+x^{c-a}}+\frac{1}{1+x^{a-b}+x^{c-b}}+\frac{1}{1+x^{a-c}+x^{b-c}}$

$=\frac{1}{1+\frac{x^b}{x^a}+\frac{x^c}{x^a}}+\frac{1}{1+\frac{x^a}{x^b}+\frac{x^c}{x^b}}+\frac{1}{1+\frac{x^a}{x^c}+\frac{x^b}{x^c}}$

$=\frac{x^a}{x^a+x^b+x^c}+\frac{x^b}{x^a+x^b+x^c}+\frac{x^c}{x^a+x^b+x^c}$

$\frac{x^a+x^b+x^c}{x^a+x^b+x^c}=1$