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Câu hỏi: Trong không gian Oxyz, cho mặt cầu $\left( S \right):{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2x-8y+9=0$ và hai điểm $A\left( 5;10;0 \right)$, $B\left( 4;2;1 \right)$. Gọi M là điểm thuộc mặt cầu $\left( S \right)$. Giá trị nhỏ nhất của $MA+3MB$ bằng
A. $\dfrac{22\sqrt{2}}{3}$.
B. $22\sqrt{2}$.
C. $11\sqrt{2}$.
D. $\dfrac{11\sqrt{2}}{3}$.
A. $\dfrac{22\sqrt{2}}{3}$.
B. $22\sqrt{2}$.
C. $11\sqrt{2}$.
D. $\dfrac{11\sqrt{2}}{3}$.
Gọi $M\left( x;y;z \right)\in \left( S \right)$.
$MA+3MB=\sqrt{{{\left( x-5 \right)}^{2}}+{{\left( y-10 \right)}^{2}}+{{z}^{2}}}+3\sqrt{{{\left( x-4 \right)}^{2}}+{{\left( y-2 \right)}^{2}}+{{\left( z-1 \right)}^{2}}}$
$=3\sqrt{{{\left( x+\dfrac{1}{2} \right)}^{2}}+{{\left( y-\dfrac{14}{3} \right)}^{2}}+{{z}^{2}}-\dfrac{8}{9}\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2x-8y+9 \right)}+3\sqrt{{{\left( x-4 \right)}^{2}}+{{\left( y-2 \right)}^{2}}+{{\left( z-1 \right)}^{2}}}$
$=3\left( \sqrt{{{\left( x+\dfrac{1}{2} \right)}^{2}}+{{\left( y-\dfrac{14}{3} \right)}^{2}}+{{z}^{2}}}+\sqrt{{{\left( x-4 \right)}^{2}}+{{\left( y-z \right)}^{2}}+{{\left( z-1 \right)}^{2}}} \right)$
$\ge \sqrt{{{\left( 4+\dfrac{1}{3} \right)}^{2}}+{{\left( 2-\dfrac{14}{3} \right)}^{2}}+{{1}^{2}}}=\dfrac{11\sqrt{2}}{3}$.
$MA+3MB=\sqrt{{{\left( x-5 \right)}^{2}}+{{\left( y-10 \right)}^{2}}+{{z}^{2}}}+3\sqrt{{{\left( x-4 \right)}^{2}}+{{\left( y-2 \right)}^{2}}+{{\left( z-1 \right)}^{2}}}$
$=3\sqrt{{{\left( x+\dfrac{1}{2} \right)}^{2}}+{{\left( y-\dfrac{14}{3} \right)}^{2}}+{{z}^{2}}-\dfrac{8}{9}\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2x-8y+9 \right)}+3\sqrt{{{\left( x-4 \right)}^{2}}+{{\left( y-2 \right)}^{2}}+{{\left( z-1 \right)}^{2}}}$
$=3\left( \sqrt{{{\left( x+\dfrac{1}{2} \right)}^{2}}+{{\left( y-\dfrac{14}{3} \right)}^{2}}+{{z}^{2}}}+\sqrt{{{\left( x-4 \right)}^{2}}+{{\left( y-z \right)}^{2}}+{{\left( z-1 \right)}^{2}}} \right)$
$\ge \sqrt{{{\left( 4+\dfrac{1}{3} \right)}^{2}}+{{\left( 2-\dfrac{14}{3} \right)}^{2}}+{{1}^{2}}}=\dfrac{11\sqrt{2}}{3}$.
Đáp án D.