The equation of the plane through the intersection of the planes x + 2y + 3z = 4 and 2x + y − z = −5 and perpendicular to the plane 5x + 3y + 6z + 8 = 0 is

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51x − 15y − 50z + 173 = 0 \text{ The equation of the plane passing through the line of intersection of the given planes is } x + 2y + 3z - 4 + \lambda \left( 2x + y - z + 5 \right) = 0 \left( 1 + 2\lambda \right)x + \left( 2 + \lambda \right)y + \left( 3 - \lambda \right)z - 4 + 5\lambda = 0 . . . \left( 1 \right) \text{ This plane is perpendicular to 5x + 3y + 6z + 8 = 0 . So } , 5 \left( 1 + 2\lambda \right) + 3\left( 2 + \lambda \right) + 6 \left( 3 - \lambda \right) = 0............... (\text{ Because } a_1 a_2 + b_1 b_2 + c_1 c_2 = 0) \Rightarrow 5 + 10\lambda + 6 + 3\lambda + 18 - 6\lambda = 0 \Rightarrow 7\lambda + 29 = 0 \Rightarrow \lambda = - $\frac{29}{7}$ \text{ Substituting this in (1), we get } \left( 1 + 2 \left( - $\frac{29}{7}$ \right) \right)x + \left( 2 + \left( - $\frac{29}{7}$ \right) \right)y + \left( 3 + $\frac{29}{7}$ \right)z - 4 + 5 \left( - $\frac{29}{7}$ \right) = 0 \Rightarrow 51x + 15y - 50z + 173 = 0

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