Trong không gian \(Oxyz\), cho \(M\left( { - 2; - 4;3} \right)\) và \(\left( P \right):2x - y + 2z - 3 = 0\), \(\left( Q \right):2x - y + 2z - 6 = 0\).

a) S, b) S, c) Đ, d) S

a) Ta có \(d\left( {M,\left( P \right)} \right) = \frac{{\left| {2.\left( { - 2} \right) + 4 + 2.3 - 3} \right|}}{{\sqrt {{2^2} + {{\left( { - 1} \right)}^2} + {2^2}} }} = 1\).

b) Ta có \(d\left( {M,\left( Q \right)} \right) = \frac{{\left| {2.\left( { - 2} \right) + 4 + 2.3 - 6} \right|}}{{\sqrt {{2^2} + {{\left( { - 1} \right)}^2} + {2^2}} }} = 0 \Rightarrow M \in \left( Q \right)\).

c) \(d\left( {\left( P \right),\left( Q \right)} \right) = d\left( {M,\left( P \right)} \right) = 1\).

d) Vì \(\left( \alpha \right)//\left( Q \right)\) nên \(\left( \alpha \right):2x - y + 2z + D = 0\).

Vì \(d\left( {\left( \alpha \right),\left( Q \right)} \right) = 2 \Leftrightarrow d\left( {M,\left( \alpha \right)} \right) = 2\)\( \Leftrightarrow \frac{{\left| {2.\left( { - 2} \right) + 4 + 2.3 + D} \right|}}{{\sqrt {{2^2} + {{\left( { - 1} \right)}^2} + {2^2}} }} = 2 \Leftrightarrow D = 0\).

Vậy \(\left( \alpha \right):2x - y + 2z = 0\).