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Question
Find the value of λ such that the line \[\frac{x - 2}{6} = \frac{y - 1}{\lambda} = \frac{z + 5}{- 4}\] is perpendicular to the plane 3x − y − 2z = 7.
Solution
\[\text{ Direction ratios of the given line are proportional to 6, } \lambda, -4.\]
\[\text{ Direction ratios of the plane are 3, -1, -2.}\]
\[\text{ Since the given line is parallel to the given plane, the line is perpendicular to the normal of the given plane. } \]
\[ \Rightarrow \left( 6 \right) \left( 3 \right) + \left( \lambda \right) \left( - 1 \right) + \left( - 4 \right) \left( - 2 \right) = 0\]
\[ \Rightarrow 18 - \lambda + 8 = 0\]
\[ \Rightarrow \lambda = 26\]
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