Question

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

Answer 1

$\left \{ {{ x^{2} +y^{2} = 20} \atop {xy=8}} \right.$
$\left \{ {{ x^{2} +y^{2} = 20} \atop {x= \frac{8}{y} }} \right.$
$\left \{ {{ \frac{64}{y ^{2} } +y^{2} = 20} \atop {x= \frac{8}{y} }} \right.$
$\left \{y^{4} -20y^{2} + 64= 0 } \atop {x= \frac{8}{y} }} \right.$
Пусть t = y². 
$t ^{2} - 20t + 64 = 0$
t₁ + t₂ = 20
t₁*t₂ = 64

t₁ = 4
t₂ = 16
Обратная замена:
y² =  4
y² = 16
$\left \{y = 2 } \atop {x= \frac{8}{y} }} \right.$
$\left \{y = -2 } \atop {x= \frac{8}{y} }} \right.$
$\left \{y = 4 } \atop {x= \frac{8}{y} }} \right.$
$\left \{y = -4 } \atop {x= \frac{8}{y} }} \right.$

$\left \{y = 2 } \atop {x= 4} }} \right.$
$\left \{y = -2 } \atop {x= -4} }} \right.$
$\left \{y = 4 } \atop {x= 2} }} \right.$
$\left \{y = -4 } \atop {x= -2} }} \right.$
ответ: $(2; 4), (-2; -4), (4; 2), (-4; -2).$

2) $\left \{ {{ x^{2} +y^{2} = 5} \atop {xy=2}} \right.$
$\left \{ {{ x^{2} +y^{2} = 5} \atop {x= \frac{2}{y} }} \right.$
$\left \{ {{ \frac{4}{y ^{2} } +y^{2} = 5} \atop {x= \frac{2}{y} }} \right.$
$\left \{y^{4} -5y^{2} + 4= 0 } \atop {x= \frac{8}{y} }} \right.$
Пусть z = y².
z² - 5z + 4 = 0

z₁ + z₂ = 5
z₁*z₂ = 4

z₁ = 4
z₂ = 1
Обратная замена:
$\left \{y = 1 } \atop {x= \frac{2}{y} }} \right.$
$\left \{y = -1 } \atop {x= \frac{2}{y} }} \right.$
$\left \{y = 2 } \atop {x= \frac{2}{y} }} \right.$
$\left \{y = -2 } \atop {x= \frac{2}{y} }} \right.$

$\left \{y = 1 } \atop {x= 2} }} \right.$
$\left \{y = -1 } \atop {x= -2} }} \right.$
$\left \{y = 2 } \atop {x= 1} }} \right.$
$\left \{y = -2 } \atop {x= -1} }} \right.$
ответ: $(1; 2), (-1; -2), (2; 1), (-2; -1).$