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x-2-y-2-4-x-2-y-2-5-find-y-x-




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

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