Tìm x;y biết: (y^2 – x^2) / 3= (x^2 – y^2) / 5 và x^10.y^10=1024 22/08/2021 Bởi Lydia Tìm x;y biết: (y^2 – x^2) / 3= (x^2 – y^2) / 5 và x^10.y^10=1024
Đáp án: $\begin{array}{l}\left\{ \begin{array}{l}\frac{{{y^2} – {x^2}}}{3} = \frac{{{x^2} – {y^2}}}{5}\\{x^{10}}.{y^{10}} = 1024 \Rightarrow {\left( {x.y} \right)^{10}} = {2^{10}} \Rightarrow xy = 2\end{array} \right.\\ \Rightarrow 5\left( {{y^2} – {x^2}} \right) – 3\left( {{x^2} – {y^2}} \right) = 0\\ \Rightarrow 5\left( {{y^2} – {x^2}} \right) + 3\left( {{y^2} – {x^2}} \right) = 0\\ \Rightarrow {y^2} – {x^2} = 0\\ \Rightarrow \left( {y – x} \right)\left( {y + x} \right) = 0\\ \Rightarrow \left[ \begin{array}{l}x = y\\x = – y\end{array} \right.\\ + )Khi\,x = y \Rightarrow x.x = 2 \Rightarrow {x^2} = 2 \Rightarrow x = \pm \sqrt 2 \\ + )\,Khi\,x = – y \Rightarrow x.\left( { – x} \right) = 2\left( {loai} \right)\\Vậy\,x = y = \pm \sqrt 2 \end{array}$ Bình luận
Đáp án:
$\begin{array}{l}
\left\{ \begin{array}{l}
\frac{{{y^2} – {x^2}}}{3} = \frac{{{x^2} – {y^2}}}{5}\\
{x^{10}}.{y^{10}} = 1024 \Rightarrow {\left( {x.y} \right)^{10}} = {2^{10}} \Rightarrow xy = 2
\end{array} \right.\\
\Rightarrow 5\left( {{y^2} – {x^2}} \right) – 3\left( {{x^2} – {y^2}} \right) = 0\\
\Rightarrow 5\left( {{y^2} – {x^2}} \right) + 3\left( {{y^2} – {x^2}} \right) = 0\\
\Rightarrow {y^2} – {x^2} = 0\\
\Rightarrow \left( {y – x} \right)\left( {y + x} \right) = 0\\
\Rightarrow \left[ \begin{array}{l}
x = y\\
x = – y
\end{array} \right.\\
+ )Khi\,x = y \Rightarrow x.x = 2 \Rightarrow {x^2} = 2 \Rightarrow x = \pm \sqrt 2 \\
+ )\,Khi\,x = – y \Rightarrow x.\left( { – x} \right) = 2\left( {loai} \right)\\
Vậy\,x = y = \pm \sqrt 2
\end{array}$
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