Question

Find the equation of the plane passing through the line of intersection of the planes 2x − y = 0 and 3z − y = 0 and perpendicular to the plane 4x + 5y − 3z = 8

Answer 1

\[\text{ The equation of the plane passing through the line of intersection of the given planes is } \]

\[2x - y + \lambda \left( 3z - y \right) = 0 \]

\[2x + \left( - 1 - \lambda \right)y + 3\lambda z = 0 . . . \left( 1 \right)\]

\[\text{ This plane is perpendicular to 4x + 5y - 3z = 8 . So },\]

\[2 \left( 4 \right) + \left( - 1 - \lambda \right) 5 - 9\lambda =\text{  0 }     (\text{ Because }  a_1 a_2 + b_1 b_2 + c_1 c_2 = 0)\]

\[ \Rightarrow 8 - 5 - 5\lambda - 9\lambda = 0\]

\[ \Rightarrow \lambda = \frac{3}{14}\]

\[\text{ Substituting this in (1), we get } \]

\[2x + \left( - 1 - \frac{3}{14} \right)y + 3\left( \frac{3}{14} \right) z = 0\]

\[ \Rightarrow 28x - 17y + 9z = 0\]