Question

If $\frac{1}{x}-\frac{1}{y}\alpha \frac{1}{x-y}$ then show that x² + y² $\alpha$ xy.

Answer 1

$$\begin{gathered} \frac{1}{x}-\frac{1}{y}\alpha \frac{1}{x-y} \\ \Rightarrow \frac{1}{x}-\frac{1}{y}=\frac{k}{x-y}\left(k\mathrm{is}\mathrm{constant}.\right) \\ \Rightarrow \frac{y-x}{xy}=\frac{k}{x-y} \\ \Rightarrow -{\left(x-y\right)}^2=kxy \\ \Rightarrow x^2+y^2-2xy=-kxy \\ \Rightarrow x^2+y^2=-kxy+2xy \\ \Rightarrow x^2+y^2=xy\left(-k+2\right) \\ \Rightarrow x^2+y^2=mxy\left(m=-k+2\mathrm{is}\mathrm{any}\mathrm{constant}.\right) \\ \therefore x^2+y^2\alpha xy \end{gathered}$$