Question

Prove that the line of section of the planes 5x + 2y − 4z + 2 = 0 and 2x + 8y + 2z − 1 = 0 is parallel to the plane 4x − 2y − 5z − 2 = 0.

 

Answer 1

\[\text{ Leta, b, c be the direction ratios of the line of section of the given planes } .\]
\[\text{ As this line lies on both the planes, their normals are perpendicular to it } .\]
\[ \Rightarrow 5a + 2b - 4c = 0 . . . \left( 1 \right)\]
\[2a + 8b + 2c = 0\]
\[ \Rightarrow a + 4b + c = 0 . . . \Rightarrow \left( 2 \right)\]
\[\text{ Using cross-multiplication method, we get } \]
\[\frac{a}{2 + 16} = \frac{b}{- 4 - 5} = \frac{c}{20 - 2}\]
\[ \Rightarrow \frac{a}{18} = \frac{b}{- 9} = \frac{c}{18}\]
\[ \Rightarrow \frac{a}{2} = \frac{b}{- 1} = \frac{c}{2}\]
\[\text{ So, the direction ratios of the line are proportional to 2, -1, 2.} \]
\[\text{ Direction ratios of the given line are 4,-2, -5.} \]
\[\text{ Now } ,\]
\[\left( 2 \right) \left( 4 \right) + \left( - 1 \right) \left( - 2 \right) + \left( 2 \right) \left( - 5 \right)\]
\[ = 8 + 2 - 10\]
\[ = 0\]
\[\text{ So, the line of section of the given planes is parallel to the given plane.} \]