Question Number 174291 by akolade last updated on 29/Jul/22
$$\mathrm{Find} \\ $$$$\frac{\mathrm{x}}{\mathrm{y}}+\frac{\mathrm{y}}{\mathrm{x}} \\ $$$$\mathrm{if}\:\mathrm{log}\left(\frac{\mathrm{x}+\mathrm{y}}{\mathrm{3}}\right)=\frac{\mathrm{logx}+\mathrm{logy}}{\mathrm{2}} \\ $$
Answered by Rasheed.Sindhi last updated on 29/Jul/22
$$\mathrm{log}\left(\frac{\mathrm{x}+\mathrm{y}}{\mathrm{3}}\right)=\frac{\mathrm{logx}+\mathrm{logy}}{\mathrm{2}}\:;\:\frac{\mathrm{x}}{\mathrm{y}}+\frac{\mathrm{y}}{\mathrm{x}}=? \\ $$$$\mathrm{log}\left(\frac{\mathrm{x}+\mathrm{y}}{\mathrm{3}}\right)=\frac{\mathrm{logxy}}{\mathrm{2}}=\mathrm{log}\left(\left(\mathrm{xy}\right)^{\mathrm{1}/\mathrm{2}} \right) \\ $$$$\frac{\mathrm{x}+\mathrm{y}}{\mathrm{3}}=\sqrt{\mathrm{xy}} \\ $$$$\left(\frac{\mathrm{x}+\mathrm{y}}{\mathrm{3}}\right)^{\mathrm{2}} =\mathrm{xy} \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{2xy}+\mathrm{y}^{\mathrm{2}} =\mathrm{9xy} \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\mathrm{7}{xy} \\ $$$$\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }{\mathrm{xy}}=\mathrm{7} \\ $$$$\frac{{x}}{{y}}+\frac{{y}}{{x}}=\mathrm{7} \\ $$