Question

Find the equation of the plane passing through the origin and perpendicular to each of the planes x + 2y − z = 1 and 3x − 4y + z = 5.

Answer 1

$$\begin{gathered} \text{The equation of any plane passing through the origin (0, 0, 0) is} \\ a\left(x-0\right)+b\left(y-0\right)+c\left(z-0\right)=0 \\ ax+by+cz=0...\left(1\right) \\ \text{It is given that (1) is perpendicular to the planes}x+2y-z=1\text{and 3}x-4y+z=5.\text{Then,} \\ a+2b-c=0...\left(2\right) \\ 3a-4b+c=0...\left(3\right) \\ \text{Solving (1), (2) and (3), we get} \\ \left|\begin{array}{ccc}x& y& z\\ 1& 2& -1\\ 3& -4& 1\end{array}\right|=0 \\ \Rightarrow -2x-4y-10z=0 \\ \Rightarrow x+2y+5z=0 \end{gathered}$$