Question

If the expression $(x+y{)}^{-1}.(x^{-1}+y^{-1}\left)\right(x{y}^{-1}+x^{-1}y{)}^{-1}$ is simplefied it takes the form of which one of the following ?

• A. x+y
• B. $(x^2+y^2{)}^{-1}$
• C. xy
• D. $x^2+y^2$

Answer 1

The correct option is B $(x^2+y^2{)}^{-1}$

${(x+y)}^{-1}{(\frac{1}{x}+\frac{1}{y})}^{-1}{(x{y}^{-1}+x^{-1}y)}^{-1}$

$\frac{1}{x+y}(\frac{1}{x}+\frac{1}{y})(\frac{x}{y}+{\frac{y}{x}}^{-1})$

=$\frac{1}{x+y}\left(\frac{x+y}{xy}\right){\left(\frac{x^2+y^2}{xy}\right)}^{-1}$

=$\frac{1}{x+y}\times \frac{x+y}{xy}\times \frac{xy}{x^2+y^2}$

=$\left(\frac{1}{x^2+y^2}\right)=(x^2+y^2{)}^{-1}$