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Câu hỏi: Trong không gian với hệ tọa độ $Oxyz$, cho hai điểm $A\left( 7 ; 9 ; 0 \right)$ ; $B\left( 0 ; 8 ; 0 \right)$ và mặt cầu $\left( S \right):{{\left( x-1 \right)}^{2}}+{{\left( y-1 \right)}^{2}}+{{z}^{2}}=25$. Với $M$ là điểm bất kì thuộc mặt cầu $\left( S \right)$, giá trị nhỏ nhất của biểu thức $P=MA+2MB$ bằng
A. $5\sqrt{2}$.
B. $\dfrac{5\sqrt{5}}{2}$.
C. $5\sqrt{5}$.
D. 10.
A. $5\sqrt{2}$.
B. $\dfrac{5\sqrt{5}}{2}$.
C. $5\sqrt{5}$.
D. 10.
Gọi $M\left( x;y;z \right)\in \left( S \right)$, ta có:
$AM=\sqrt{{{\left( x-7 \right)}^{2}}+{{\left( y-9 \right)}^{2}}+{{z}^{2}}}$, $BM=\sqrt{{{x}^{2}}+{{\left( y-8 \right)}^{2}}+{{z}^{2}}}$.
Vì $M\in \left( S \right)$ nên $T={{\left( x-1 \right)}^{2}}+{{\left( y-1 \right)}^{2}}+{{z}^{2}}-25=0$.
$\Rightarrow P=MA+2MB=\sqrt{{{\left( x-7 \right)}^{2}}+{{\left( y-9 \right)}^{2}}+{{z}^{2}}+3T}+2\sqrt{{{x}^{2}}+{{\left( y-8 \right)}^{2}}+{{z}^{2}}}$ $=\sqrt{{{\left( x-7 \right)}^{2}}+{{\left( y-9 \right)}^{2}}+{{z}^{2}}+3{{\left( x-1 \right)}^{2}}+3{{\left( y-1 \right)}^{2}}+3{{z}^{2}}-75}+2\sqrt{{{x}^{2}}+{{\left( y-8 \right)}^{2}}+{{z}^{2}}}$
$=\sqrt{4{{x}^{2}}+4{{y}^{2}}+4{{z}^{2}}-20x-24y+61}+2\sqrt{{{x}^{2}}+{{\left( y-8 \right)}^{2}}+{{z}^{2}}}$
$=2\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-5x-6y+\dfrac{61}{4}}-2\sqrt{{{x}^{2}}+{{\left( y-8 \right)}^{2}}+{{z}^{2}}}$.
$=2\sqrt{{{\left( x-\dfrac{5}{2} \right)}^{2}}+{{\left( y-3 \right)}^{2}}+{{z}^{2}}}+2\sqrt{{{x}^{2}}+{{\left( y-8 \right)}^{2}}+{{z}^{2}}}$.
Đặt $\overrightarrow{u}=\left( x-\dfrac{5}{2};y-3;z \right); \overrightarrow{v}=\left( -x;8-y;-z \right)$ $\Rightarrow P=2\left| \overrightarrow{u} \right|+2\left| \overrightarrow{v} \right|\ge 2\left| \overrightarrow{u}+\overrightarrow{v} \right|=2\sqrt{{{\left( \dfrac{5}{2} \right)}^{2}}+{{5}^{2}}}=5\sqrt{5}$.
Do đó $\min P=5\sqrt{5}\Leftrightarrow M\left( 1;6;0 \right)$.
$AM=\sqrt{{{\left( x-7 \right)}^{2}}+{{\left( y-9 \right)}^{2}}+{{z}^{2}}}$, $BM=\sqrt{{{x}^{2}}+{{\left( y-8 \right)}^{2}}+{{z}^{2}}}$.
Vì $M\in \left( S \right)$ nên $T={{\left( x-1 \right)}^{2}}+{{\left( y-1 \right)}^{2}}+{{z}^{2}}-25=0$.
$\Rightarrow P=MA+2MB=\sqrt{{{\left( x-7 \right)}^{2}}+{{\left( y-9 \right)}^{2}}+{{z}^{2}}+3T}+2\sqrt{{{x}^{2}}+{{\left( y-8 \right)}^{2}}+{{z}^{2}}}$ $=\sqrt{{{\left( x-7 \right)}^{2}}+{{\left( y-9 \right)}^{2}}+{{z}^{2}}+3{{\left( x-1 \right)}^{2}}+3{{\left( y-1 \right)}^{2}}+3{{z}^{2}}-75}+2\sqrt{{{x}^{2}}+{{\left( y-8 \right)}^{2}}+{{z}^{2}}}$
$=\sqrt{4{{x}^{2}}+4{{y}^{2}}+4{{z}^{2}}-20x-24y+61}+2\sqrt{{{x}^{2}}+{{\left( y-8 \right)}^{2}}+{{z}^{2}}}$
$=2\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-5x-6y+\dfrac{61}{4}}-2\sqrt{{{x}^{2}}+{{\left( y-8 \right)}^{2}}+{{z}^{2}}}$.
$=2\sqrt{{{\left( x-\dfrac{5}{2} \right)}^{2}}+{{\left( y-3 \right)}^{2}}+{{z}^{2}}}+2\sqrt{{{x}^{2}}+{{\left( y-8 \right)}^{2}}+{{z}^{2}}}$.
Đặt $\overrightarrow{u}=\left( x-\dfrac{5}{2};y-3;z \right); \overrightarrow{v}=\left( -x;8-y;-z \right)$ $\Rightarrow P=2\left| \overrightarrow{u} \right|+2\left| \overrightarrow{v} \right|\ge 2\left| \overrightarrow{u}+\overrightarrow{v} \right|=2\sqrt{{{\left( \dfrac{5}{2} \right)}^{2}}+{{5}^{2}}}=5\sqrt{5}$.
Do đó $\min P=5\sqrt{5}\Leftrightarrow M\left( 1;6;0 \right)$.
Đáp án C.