According to ques,
planes are
\[{{P}_{1}}:\text{ }5x\text{ }+\text{ }3y\text{ }+\text{ }6z\text{ }+\text{ }8\text{ }=\text{ }0\]
\[{{P}_{2}}:\text{ }x\text{ }+\text{ }2y\text{ }+\text{ }3z\text{ }\text{ }4\text{ }=\text{ }0\]
\[{{P}_{3}}:\text{ }2x\text{ }+\text{ }y\text{ }\text{ }z\text{ }+\text{ }5\text{ }=\text{ }0\]
So,
the equation of the plane passing through the line of intersection of P1 and P3 is:
\[\left( x\text{ }+\text{ }2y\text{ }+\text{ }3z\text{ }\text{ }4 \right)\text{ }+\text{ }\lambda \left( 2x\text{ }+\text{ }y\text{ }\text{ }z\text{ }+\text{ }5 \right)\text{ }=\text{ }0\]
\[\left( 1\text{ }+\text{ }2\lambda \right)x\text{ }+\text{ }\left( 2\text{ }+\text{ }\lambda \right)y\text{ }+\text{ }\left( 3\text{ }\text{ }\lambda \right)z\text{ }\text{ }4\text{ }+\text{ }5\lambda \text{ }=\text{ }0\text{ }\ldots .\text{ }\left( i \right)\]
Since plane (i) is perpendicular to P1,
\[5\left( 1\text{ }+\text{ }2\lambda \right)\text{ }+\text{ }3\left( 2\text{ }+\text{ }\lambda \right)\text{ }+\text{ }6\left( 3\text{ }\text{ }\lambda \right)\text{ }=\text{ }0\]
\[5\text{ }+\text{ }10\lambda \text{ }+\text{ }6\text{ }+\text{ }3\lambda \text{ }+\text{ }18\text{ }\text{ }6\lambda \text{ }=\text{ }0\]
\[7\lambda \text{ }+\text{ }29\text{ }=\text{ }0\]
\[\lambda \text{ }=\text{ }-29/7\]
Putting the values in equation (i),
\[-15x\text{ }\text{ }15y\text{ }+\text{ }50z\text{ }\text{ }28\text{ }\text{ }145\text{ }=\text{ }0\]
\[-15x\text{ }\text{ }15y\text{ }+\text{ }50z\text{ }\text{ }173\text{ }=\text{ }0\Rightarrow 51x\text{ }+\text{ }15y\text{ }\text{ }50z\text{ }+\text{ }173\text{ }=\text{ }0\]
Therefore,
the required equation of plane is:
\[~51x\text{ }+\text{ }15y\text{ }\text{ }50z\text{ }+\text{ }173\text{ }=\text{ }0.\]