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