Gọi \(\overrightarrow{n}=\left(A;B;C\right) \left(A^{2} +B^{2} +C^{2} \ne 0\right)\) là véc tơ pháp tuyến của \(\left(Q\right) \Leftrightarrow \overrightarrow{n}.\overrightarrow{MN}=0 \Leftrightarrow -2A-B+2C=0 \Leftrightarrow B=2C-2A \)
Phương trình mặt phẳng\(\left(Q\right):A\left(x+1\right)+B\left(y-1\right)+C\left(z-1\right)=0\Leftrightarrow Ax+By+Cz+A-B-C=0 \)
\(\Rightarrow d\left(E,\left(Q\right)\right)=\frac{\left|2A-B\right|}{\sqrt{A^{2} +B^{2} +C^{2} } } =\frac{\left|4A-2C\right|}{\sqrt{5A^{2} -8AC+5C^{2} } } =\frac{2}{\sqrt{26} } \)
\(\Leftrightarrow \frac{\left|2A-C\right|}{\sqrt{5A^{2} -8AC+5C^{2} } } =\frac{1}{\sqrt{26} } \)
\(\Leftrightarrow 99A^{2} -96AC+21C^{2} =0\Leftrightarrow \left[\begin{array}{l} {A=\frac{7}{11} C} \\ {A=\frac{1}{3} C} \end{array}\right. \)
* Với \(A=\frac{7}{11} C\Rightarrow \overrightarrow{n}=\left(7;8;11\right)\Rightarrow \left(Q\right):7x+8y+11z-12=0 \)
* Với \(A=\frac{1}{3} C\Rightarrow \overrightarrow{n}=\left(1;4;3\right)\Rightarrow \left(Q\right):x+4y+3z-6=0 \)