Question

Sabe-se que $\mathsf{\frac{x + y}{x - y} = \sqrt{2}}$. Determine o valor de $\mathsf{\frac{x^2 + y^2}{xy}}$.

Answer 1

Olá DanJR,


Elevando os dois lados da equação ao quadrado temos:

$\mathsf{\Big(\dfrac{x+y}{x-y}\Big)^2=\big(\sqrt{2}~\big)^2\Rightarrow \dfrac{x^2+xy+xy+y^2}{x^2-xy-xy+y^2}=2\Rightarrow\dfrac{x^2+2xy+y^2}{x^2-2xy+y^2}=2}\\\\=\\\\\mathsf{x^2+2xy+y^2=2\cdot(x^2-2xy+y^2)\Rightarrow x^2+2xy+y^2=2x^2-4xy+2y^2}\\\\=\\\\\mathsf{2x^2-x^2+2y^2-y^2=2xy+4xy\Rightarrow x^2+y^2=6xy}\\\\=\\\\\boxed{\mathsf{\dfrac{x^2+y^2}{xy}=6}}$