\n
Problema<\/p>\n
\nSabe-se que
\n[tex]\\qquad\\qquad \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\dfrac{x + y}{x \u2212 y}+\\dfrac{x \u2212 y}{x + y}= 3[\/tex].<\/p>\n
Determine o valor de
\n[tex]\\qquad\\qquad \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\dfrac{x^2 + y^2}{x^2 \u2212 y^2}+\\dfrac{x^2 \u2212 y^2}{x^2 + y^2}[\/tex].<\/p>\n
\n<\/div>\n
\n
\n
Solu\u00e7\u00e3o<\/p>\n
\nObserve que:
\n[tex]\\qquad \\begin{align*} 3= \\dfrac{x + y}{x \u2212 y}+\\dfrac{x \u2212 y}{x + y} & = \\dfrac{\\left(x + y\\right)^2+\\left(x \u2212 y\\right)^2}{x^2 \u2212 y^2}
\n\\\\&=\\dfrac{\\left(x^2+2xy+y^2\\right)+\\left(x^2 \u2212 2xy + y^2\\right)}{x^2 \u2212 y^2}
\n\\\\& =\\dfrac{2\\left(x^2+y^2\\right)}{x^2 \u2212 y^2}.
\n\\end{align*}[\/tex]
\nAssim
\n[tex] \\qquad \\dfrac{x^2+y^2}{x^2 \u2212 y^2}=\\dfrac{3}{2}[\/tex]
\ne
\n[tex] \\qquad \\dfrac{x^2-y^2}{x^2 + y^2}=\\dfrac{2}{3}[\/tex],
\ndonde
\n[tex]\\qquad \\dfrac{x^2 + y^2}{x^2 \u2212 y^2}+\\dfrac{x^2 \u2212 y^2}{x^2 + y^2}= \\dfrac{3}{2}+\\dfrac{2}{3}[\/tex].
\nDessa forma,
\n[tex]\\qquad \\boxed{\\dfrac{x^2 + y^2}{x^2 \u2212 y^2}+\\dfrac{x^2 \u2212 y^2}{x^2 + y^2}= \\dfrac{13}{6}}[\/tex].<\/p>\n
\n
Solu\u00e7\u00e3o elaborada pelos Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n\n
\nPrimeira Gincana de 2016 – Clubes de Matem\u00e1tica da OBMEP.
\nN\u00edvel C – Quest\u00e3o M\u00e9dia.
\nO COM
Paralelo 38<\/strong> participou da discuss\u00e3o da quest\u00e3o no Blog.<\/div>\n","protected":false},"excerpt":{"rendered":"Problema Sabe-se que [tex]\\qquad\\qquad \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\dfrac{x + y}{x \u2212 y}+\\dfrac{x \u2212 y}{x + y}= 3[\/tex]. Determine o valor de [tex]\\qquad\\qquad \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\dfrac{x^2 + y^2}{x^2 \u2212 y^2}+\\dfrac{x^2 \u2212 y^2}{x^2 … <\/p>\n
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