Система уравнений решите 1)x^2+y^2=20 xy=8 2)x^2+y^2=5 xy=2

[latex] \left \{ {{ x^{2} +y^{2} = 20} \atop {xy=8}} \right. [/latex] [latex] \left \{ {{ x^{2} +y^{2} = 20} \atop {x= \frac{8}{y} }} \right. [/latex] [latex] \left \{ {{ \frac{64}{y ^{2} } +y^{2} = 20} \atop {x= \frac{8}{y} }} \right. [/latex] [latex] \left \{y^{4} -20y^{2} + 64= 0 } \atop {x= \frac{8}{y} }} \right. [/latex] Пусть t = y².  [latex]t ^{2} - 20t + 64 = 0[/latex] t₁ + t₂ = 20 t₁*t₂ = 64 t₁ = 4 t₂ = 16 Обратная замена: y² =  4 y² = 16 [latex]\left \{y = 2 } \atop {x= \frac{8}{y} }} \right.[/latex] [latex]\left \{y = -2 } \atop {x= \frac{8}{y} }} \right.[/latex] [latex]\left \{y = 4 } \atop {x= \frac{8}{y} }} \right.[/latex] [latex]\left \{y = -4 } \atop {x= \frac{8}{y} }} \right.[/latex] [latex]\left \{y = 2 } \atop {x= 4} }} \right.[/latex] [latex]\left \{y = -2 } \atop {x= -4} }} \right.[/latex] [latex]\left \{y = 4 } \atop {x= 2} }} \right.[/latex] [latex]\left \{y = -4 } \atop {x= -2} }} \right.[/latex] Ответ: [latex](2; 4), (-2; -4), (4; 2), (-4; -2).[/latex] 2) [latex]\left \{ {{ x^{2} +y^{2} = 5} \atop {xy=2}} \right. [/latex] [latex]\left \{ {{ x^{2} +y^{2} = 5} \atop {x= \frac{2}{y} }} \right.[/latex] [latex]\left \{ {{ \frac{4}{y ^{2} } +y^{2} = 5} \atop {x= \frac{2}{y} }} \right.[/latex] [latex]\left \{y^{4} -5y^{2} + 4= 0 } \atop {x= \frac{8}{y} }} \right.[/latex] Пусть z = y². z² - 5z + 4 = 0 z₁ + z₂ = 5 z₁*z₂ = 4 z₁ = 4 z₂ = 1 Обратная замена: [latex]\left \{y = 1 } \atop {x= \frac{2}{y} }} \right.[/latex] [latex]\left \{y = -1 } \atop {x= \frac{2}{y} }} \right.[/latex] [latex]\left \{y = 2 } \atop {x= \frac{2}{y} }} \right.[/latex] [latex]\left \{y = -2 } \atop {x= \frac{2}{y} }} \right.[/latex] [latex]\left \{y = 1 } \atop {x= 2} }} \right.[/latex] [latex]\left \{y = -1 } \atop {x= -2} }} \right.[/latex] [latex]\left \{y = 2 } \atop {x= 1} }} \right.[/latex] [latex]\left \{y = -2 } \atop {x= -1} }} \right.[/latex] Ответ: [latex](1; 2), (-1; -2), (2; 1), (-2; -1).[/latex]