Question

sabendo-se que x²+y²/(x+y)² = 7/12, qual o valor de x/y + y/x ?

Answer 1

$\dfrac{x^2+y^2}{(x+y)^2}=\dfrac{7}{12}\ \to\ \dfrac{(x+y)^2-2xy}{(x+y)^2}=\dfrac{7}{12}\ \to\ 1-\dfrac{2xy}{(x+y)^2}=\dfrac{7}{12}\ \to\\\\ \to\ \dfrac{2xy}{(x+y)^2}=\dfrac{12}{12}-\dfrac{7}{12}\ \to\ \dfrac{x^2+y^2+2xy}{2xy}=\dfrac{12}{5}\ \to\\\\ \to\ \dfrac{x}{2y}+\dfrac{y}{2x}+1=\dfrac{12}{5}\ \to\ \dfrac{1}{2}\bigg(\dfrac{x}{y}+\dfrac{y}{x}\bigg)=\dfrac{12}{5}-\dfrac{5}{5}\ \to\\\\ \to\ \dfrac{x}{y}+\dfrac{y}{x}=2\bigg(\dfrac{7}{5}\bigg) \to\ \boxed{\dfrac{x}{y}+\dfrac{y}{x}=\dfrac{14}{5}}$