Question

Find the equation of the plane mid-parallel to the planes 2x − 2y + z + 3 = 0 and 2x − 2y + z + 9 = 0.

Answer 1

$$\begin{gathered} \text{We know that the distance between two planes}ax+by+cz=d_1\text{and}ax+by+cz=d_2\text{is}\frac{\left|d_2-d_1\right|}{\sqrt{a^2+b^2+c^2}} \\ \text{The equation of plane that is mid-parallel to the planes} \\ 2x-2y+z+3=0...\left(1\right) \\ 2x-2y+z+9=0...\left(2\right) \\ \text{is of the form}2x-2y+z+k=0...\left(3\right) \\ \text{It means that the distance between (1) and (3) = distance between (1) and (2)} \\ \Rightarrow \frac{\left|k-3\right|}{\sqrt{4+4+1}}=\frac{\left|k-9\right|}{\sqrt{4+4+1}} \\ \Rightarrow \left|k-3\right|=\left|k-9\right| \\ \Rightarrow k-3=k-9\text{or}k-3=-\left(k-9\right) \\ \Rightarrow 3=9\text{(false);}k-3=-k+9 \\ \Rightarrow 2k=12 \\ \Rightarrow k=6 \\ \text{Substituting this in (3), we get} \\ 2x-2y+z+6=0,\text{which is the required equation of the plane.} \end{gathered}$$