If ab + c = bc + a = ca + b = k, then find the value of k is

Answer: Option D

$$\begin{array}{rl}& \frac{a}{b+c}=k\\ & \Rightarrow a=k\left(b+c\right)....(i)\\ & \frac{b}{c+a}=k\\ & \Rightarrow b=k\left(c+a\right)....(ii)\\ & \frac{c}{a+b}=k\\ & \Rightarrow c=k\left(a+b\right)....(iii)\\ & \text{Adding (i), (ii) and (iii) we get,}\\ & a+b+c\\ & =k\left(2a+2b+2c\right)\\ & \Rightarrow k=\frac{a+b+c}{2(a+b+c)}\\ & =\frac{1}{2}\end{array}$$