Question

Find the equation of the plane through the line of intersection of the planes x + 2y + 3z + 4 = 0 and x − y + z + 3 = 0 and passing through the origin.

Answer 1

$$\begin{gathered} \text{The equation of the plane passing through the line of intersection of the given planes is} \\ x+2y+3z+4+\lambda \left(x-y+z+3\right)=0...\left(1\right) \\ \text{This passes through (0, 0, 0). So,} \\ 0+0+0+4+\lambda \left(0-0+0+3\right)=0 \\ \Rightarrow 4+3\lambda =0 \\ \Rightarrow \lambda =\frac{-4}{3} \\ \text{Substituting this in (1), we get} \\ x+2y+3z+4-\frac{4}{3}\left(x-y+z+3\right)=0 \\ \Rightarrow -x+10y+5z=0 \\ \Rightarrow x-10y-5z=0 \end{gathered}$$