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

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