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Question
Write the value of k for which the planes x − 2y + kz = 4 and 2x + 5y − z = 9 are perpendicular.
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Solution
\[\text{ We know that the planes } a_1 x + b_1 y + c_1 z + d_1 = 0 \text{ and } a_2 x + b_2 y + c_2 z + d_2 = 0 \text{ are perpendicular to each other only if} \]
\[ a_1 a_2 + b_1 b_2 + c_1 c_2 = 0\]
\[\text{ The given planes are x - 2y + kz = 4 and } 2x + 5y - z = 9\]
\[ \Rightarrow a_1 = 1; b_1 = - 2; c_1 = k; a_2 = 2; b_2 = 5; c_2 = - 1\]
\[\text{ It is given that the given planes are perpendicular } .\]
\[ \Rightarrow a_1 a_2 + b_1 b_2 + c_1 c_2 = 0\]
\[ \Rightarrow \left( 1 \right) \left( 2 \right) + \left( - 2 \right) \left( 5 \right) + \left( k \right) \left( - 1 \right) = 0\]
\[ \Rightarrow 2 - 10 - k = 0\]
\[ \Rightarrow - 8 - k = 0\]
\[ \Rightarrow k = - 8\]
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