Question

Решить уравнение {x/y+y/x=5,2 {x^2-y^2=24

Answer 1

Преобразуем первое уравнение, сделав замену :

$\frac{x}{y}=a\\\\\frac{y}{x}=\frac{1}{a}\\\\a+\frac{1}{a}=5,2\\\\a^{2}-5,2a+1=0,a\neq0\\\\D=(-5,2)^{2}-4*1=27,04-4=23,04=4,8^{2}\\\\a_{1}=\frac{5,2+4,8}{2}=5\\\\a_{2} =\frac{5,2-4,8}{2}=0,2=\frac{1}{5}$

$1)\frac{x}{y}=5\\\\x=5y\\\\\left \{ {{x=5y} \atop {x^{2}-y^{2}=24}} \right.\\\\\left \{ {{x=5y} \atop {25y^{2}-y^{2}=24}} \right.\\\\\left \{ {{x=5y} \atop {24y^{2}=24 }} \right.\\\\\left \{ {{x=5y} \atop {y^{2}=1 }} \right.\\\\y_{1} =1;x_{1}=5\\\\y_{2} =-1;x_{2}=-5$

$2)\frac{x}{y}=\frac{1}{5}\\\\y=5x\\\\\left \{ {{y=5x} \atop {x^{2}-25x^{2} =24 }} \right.\\\\\left \{ {{y=5x} \atop {-24x^{2}=24 }} \right.\\\\\left \{ {{y=5x} \atop {x^{2}=-1<0 }} \right.$

Решений нет

ответ : (5 ; 1) , (- 5 ; - 1)