If x + 1x - 1 = a/b and 1 - y1 + y = b/a , then the value of x - y1 + xy = ?

Answer: Option A

$$\begin{array}{rl}& \frac{x+1}{x-1}\text{ = }\frac{a}{b}\\ & \text{By componendo and dividendo}\\ & \frac{x+1+x-1}{x+1-x+1}=\frac{a+b}{a-b}\\ & \Rightarrow \frac{2x}{2}=\frac{a+b}{a-b}\\ & \Rightarrow x=\frac{a+b}{a-b}.....(i)\\ & \text{Again ,}\frac{1-y}{1+y}\text{ = }\frac{b}{a}\\ & \Rightarrow \frac{1+y}{1-y}=\frac{a}{b}\\ & \Rightarrow \frac{1+y+1-y}{1+y-1+y}=\frac{a+b}{a-b}\\ & \Rightarrow \frac{2}{2y}=\frac{a+b}{a-b}\\ & \Rightarrow \frac{1}{y}=\frac{a+b}{a-b}\\ & \Rightarrow y=\frac{a-b}{a+b}.....(ii)\\ & \text{Subtracting equation (ii) from (i) we get }\\ & \therefore x-y=\frac{a+b}{a-b}-\frac{a-b}{a+b}\\ & \left[\because {\left(a+b\right)}^2-{\left(a-b\right)}^2=4aband{a}^2-b^2=\left(a-b\right)\left(a+b\right)\right]\\ & =\frac{{\left(a+b\right)}^2-{\left(a-b\right)}^2}{\left(a-b\right)\left(a+b\right)}\\ & =\frac{4ab}{a^2-b^2}\\ & \text{Multiply equation (i) and (ii) we get}\\ & xy=\frac{a+b}{a-b}\times \frac{a-b}{a+b}=1\\ & \therefore \text{Expression,}\\ & \text{ = }\frac{x-y}{1+xy}=\frac{\frac{4ab}{a^2-b^2}}{1+1}\\ & =\frac{4ab}{2\left(a^2-b^2\right)}\\ & =\frac{2ab}{a^2-b^2}\end{array}$$