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

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