\(\dfrac{x^3}{8}=\dfrac{y^3}{64}=\dfrac{z^3}{216}\)
\(\Leftrightarrow\left(\dfrac{x}{2}\right)^3=\left(\dfrac{y}{4}\right)^3=\left(\dfrac{z}{6}\right)^3\)
\(\Leftrightarrow\dfrac{x}{2}=\dfrac{y}{4}=\dfrac{z}{6}\)
Đặt : \(\dfrac{x}{2}=\dfrac{y}{4}=\dfrac{z}{6}=k\) \(\Leftrightarrow\left\{{}\begin{matrix}x=2k\\y=4k\\z=6k\end{matrix}\right.\)
Mà \(2x^2+2y^2-z^2=1\)
\(\Leftrightarrow2.\left(2k\right)^2+2.\left(4k\right)^2-\left(6k\right)^2=1\)
\(\Leftrightarrow8k^2+32k^2-36k^2=1\)
\(\Leftrightarrow4k^2=1\)
\(\Leftrightarrow k^2=\dfrac{1}{4}\) \(\Leftrightarrow\left[{}\begin{matrix}k=\dfrac{1}{2}\\k=-\dfrac{1}{2}\end{matrix}\right.\)
+) \(k=\dfrac{1}{2}\) \(\Leftrightarrow\left\{{}\begin{matrix}x=2.\dfrac{1}{2}=1\\y=4.\dfrac{1}{2}=2\\z=6.\dfrac{1}{2}=3\end{matrix}\right.\)
+) \(k=-\dfrac{1}{2}\) \(\Leftrightarrow\left\{{}\begin{matrix}x=-\dfrac{1}{2}.2=-1\\y=-\dfrac{1}{2}.4=-2\\z=-\dfrac{1}{2}.6=-3\end{matrix}\right.\)
