The equation of the plane through the line x + y + z + 3 = 0 = 2x − y + 3z + 1 and parallel to the line x/1 = y/2 = z/3 is

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x − 5y + 3z = 7 \text{ The equation of the plane passing through the line of intersection of the given planes is } x + y + z + 3 + \lambda \left( 2x - y + 3z + 1 \right) = 0 \left( 1 + 2\lambda \right)x + \left( 1 - \lambda \right)y + \left( 1 + 3\lambda \right)z + 3 + \lambda = 0 . . . \left( 1 \right) \text{ This plane is parallel to the line} x/1=y/2=z/3 . \text{ It means that this line is perpendicular to the normal of the plane (1) } . \Rightarrow 1 \left( 1 + 2\lambda \right) + 2 \left( 1 - \lambda \right) + 3 \left( 1 + 3\lambda \right) = 0.................... (\text{ Because }a_1 a_2 + b_1 b_2 + c_1 c_2 = 0) \Rightarrow 1 + 2\lambda + 2 - 2\lambda + 3 + 9\lambda = 0 \Rightarrow 9\lambda + 6 = 0 \Rightarrow \lambda = - $\frac{2}{3}$ \text{ Substituting this in (1), we get } \left( 1 + 2 \left( - $\frac{2}{3}$ \right) \right)x + \left( 1 - \left( - $\frac{2}{3}$ \right) \right)y + \left( 1 + 3 \left( - $\frac{2}{3}$ \right) \right)z + 3 + \left( - $\frac{2}{3}$ \right) = 0 \Rightarrow - x + 5y - 3z + 7 = 0 \Rightarrow x - 5y + 3z = 7

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