Toán Lớp 9: x căn (1-y^2)+y căn (1-z^2)+ z căn (1-x^2)=3/2
tính x^2+y^2+z^2
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Toán Lớp 9: x căn (1-y^2)+y căn (1-z^2)+ z căn (1-x^2)=3/2
tính x^2+y^2+z^2
Comments ( 2 )
x\sqrt {1 – {y^2}} + y\sqrt {1 – {z^2}} + z\sqrt {1 – {x^2}} = \dfrac{3}{2}\\
\Leftrightarrow 2x\sqrt {1 – {y^2}} + 2y\sqrt {1 – {z^2}} + 2z\sqrt {1 – {x^2}} = 3\\
\Leftrightarrow 3 – 2x\sqrt {1 – {y^2}} + 2y\sqrt {1 – {z^2}} + 2z\sqrt {1 – {x^2}} = 0\\
\Leftrightarrow \left( {{x^2} – 2x\sqrt {1 – {y^2}} + 1 – {y^2}} \right) + \left( {{y^2} – 2y\sqrt {1 – {z^2}} + 1 – {z^2}} \right)\\
+ \left( {{z^2} – 2z\sqrt {1 – {x^2}} + 1 – {x^2}} \right) = 0\\
\Leftrightarrow {\left( {x – \sqrt {1 – {y^2}} } \right)^2} + {\left( {y – \sqrt {1 – {z^2}} } \right)^2} + {\left( {z – \sqrt {1 – {x^2}} } \right)^2} = 0\\
\Leftrightarrow \left\{ \begin{array}{l}
x = \sqrt {1 – {y^2}} \\
y = \sqrt {1 – {z^2}} \\
z = \sqrt {1 – {x^2}}
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
{x^2} = 1 – {y^2}\\
{y^2} = 1 – {z^2}\\
{z^2} = 1 – {x^2}
\end{array} \right.\\
\Rightarrow 2\left( {{x^2} + {y^2} + {z^2}} \right) = 3\\
\Rightarrow {x^2} + {y^2} + {z^2} = \dfrac{3}{2}
\end{array}$