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