Question Number 201302 by hardmath last updated on 03/Dec/23
$$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{−\mathrm{2}} \:=\:\mathrm{4}}\\{\mathrm{x}^{−\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:=\:\mathrm{5}}\end{cases}\:\:\:\:\:\mathrm{find}:\:\:\frac{\mathrm{y}}{\mathrm{x}}\:=\:? \\ $$
Answered by witcher3 last updated on 03/Dec/23
$$\left(\mathrm{1}\right)\ast\left(\mathrm{2}\right)\Leftrightarrow\mathrm{2}+\left(\mathrm{xy}\right)^{\mathrm{2}} +\frac{\mathrm{1}}{\left(\mathrm{xy}\right)^{\mathrm{2}} }=\mathrm{20} \\ $$$$\left(\mathrm{2}\right)/\left(\mathrm{1}\right)\Rightarrow\frac{\mathrm{1}+\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} }.\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} +\mathrm{1}}=\frac{\mathrm{5}}{\mathrm{4}} \\ $$$$\frac{\mathrm{y}}{\mathrm{x}}=\underset{−} {+}\sqrt{\frac{\mathrm{5}}{\mathrm{4}}} \\ $$