Question

If x² + y² α xy show that x α y.

Answer 1

$$\begin{gathered} \mathrm{If}{x}^2+y^2\alpha xy \\ \Rightarrow x^2+y^2=kxy\left(k\mathrm{is}\mathrm{constant}\right) \\ \Rightarrow \frac{x^2+y^2}{xy}=k \\ \Rightarrow \frac{x}{y}+\frac{y}{x}=k \\ \mathrm{Let}\frac{x}{y}=z \\ \therefore z+\frac{1}{z}=k \\ \Rightarrow z^2+1=kz \\ \Rightarrow z^2-kz+1=0 \\ \Rightarrow z=\frac{k\pm \sqrt{k^2-4}}{2} \\ Let\frac{k+\sqrt{k^2-4}}{2}=C\mathrm{and}\frac{k-\sqrt{k^2-4}}{2}=C\text{'} \\ \therefore z=C\mathrm{or}C\text{'} \\ \Rightarrow \frac{x}{y}=C\mathrm{or}\frac{x}{y}=C\text{'} \\ \Rightarrow x=Cy\mathrm{or}x=C\text{'}y \\ \Rightarrow x\mathrm{\alpha }y \end{gathered}$$