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Giả sử mặt phẳng (P) có dạng:
$\begin{array}{l}
Ax + By + Cz + D = 0\,\,({A^2} + {B^2} + {C^2} \ne 0) \Rightarrow \overrightarrow {{n_P}} = (A;B;C).\\
M \in (P) \Rightarrow 3B - 2C + D = 0\,\,(1)\\
(P)//d
\Rightarrow \overrightarrow{n_{(P)} } . \overrightarrow{u_d}=0 \Rightarrow {\rm A} + {\rm B} + 4C = 0\,\,(2)\\
d(d,(P)) = 3 \Leftrightarrow \frac{{|C + D|}}{{\sqrt {{A^2} + {B^2} + {C^2}} }} = 3\\
\Rightarrow \left[ \begin{array}{l}
B = - 2C\\
B = - 8C
\end{array} \right.
\end{array}$
TH1: $B = - 2C$, chọn $C = - 1,B = 2 \Rightarrow {\rm A} = 2,D = - 8 $
$\Rightarrow (P):2x + 2y - z - 8 = 0$
TH2: $B = - 8C$, chọn $C = 1,B = - 8 \Rightarrow {\rm A} = 4,D = 26 $
$\Rightarrow (P):4x - 8y + z + 26 = 0$
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