According to ques,
planes are
![\[{{P}_{1}}:\text{ }5x\text{ }+\text{ }3y\text{ }+\text{ }6z\text{ }+\text{ }8\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2b74eaa1b948ff4bcbf32c10e8198dff_l3.png "Rendered by QuickLaTeX.com")
![\[{{P}_{2}}:\text{ }x\text{ }+\text{ }2y\text{ }+\text{ }3z\text{ }\text{ }4\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1ef633bc9e593a8e3efe526286bb985b_l3.png "Rendered by QuickLaTeX.com")
![\[{{P}_{3}}:\text{ }2x\text{ }+\text{ }y\text{ }\text{ }z\text{ }+\text{ }5\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-e6c56c61a781c6aff7228fd9d7d44a23_l3.png "Rendered by QuickLaTeX.com")
So,
the equation of the plane passing through the line of intersection of P1 and P3 is:
![\[\left( x\text{ }+\text{ }2y\text{ }+\text{ }3z\text{ }\text{ }4 \right)\text{ }+\text{ }\lambda \left( 2x\text{ }+\text{ }y\text{ }\text{ }z\text{ }+\text{ }5 \right)\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-26667feca0468eda9c0648b0bc57e34f_l3.png "Rendered by QuickLaTeX.com")
![\[\left( 1\text{ }+\text{ }2\lambda \right)x\text{ }+\text{ }\left( 2\text{ }+\text{ }\lambda \right)y\text{ }+\text{ }\left( 3\text{ }\text{ }\lambda \right)z\text{ }\text{ }4\text{ }+\text{ }5\lambda \text{ }=\text{ }0\text{ }\ldots .\text{ }\left( i \right)\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-06644c20b7bb786a9d54245f78cb45d7_l3.png "Rendered by QuickLaTeX.com")
Since plane (i) is perpendicular to P1,
![\[5\left( 1\text{ }+\text{ }2\lambda \right)\text{ }+\text{ }3\left( 2\text{ }+\text{ }\lambda \right)\text{ }+\text{ }6\left( 3\text{ }\text{ }\lambda \right)\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ef6794e699fc213c86c954435cc0ca19_l3.png "Rendered by QuickLaTeX.com")
![\[5\text{ }+\text{ }10\lambda \text{ }+\text{ }6\text{ }+\text{ }3\lambda \text{ }+\text{ }18\text{ }\text{ }6\lambda \text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-423a292974eaa57c094712d06568360f_l3.png "Rendered by QuickLaTeX.com")
![\[7\lambda \text{ }+\text{ }29\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1344d9c8ec3ca40c0141351460268c78_l3.png "Rendered by QuickLaTeX.com")
![\[\lambda \text{ }=\text{ }-29/7\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-4de46b83a35472ccaf92a19fd7eac305_l3.png "Rendered by QuickLaTeX.com")
Putting the values in equation (i),
![\[-15x\text{ }\text{ }15y\text{ }+\text{ }50z\text{ }\text{ }28\text{ }\text{ }145\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ea99c810f8c8936544e6566fb4c5aabf_l3.png "Rendered by QuickLaTeX.com")
![\[-15x\text{ }\text{ }15y\text{ }+\text{ }50z\text{ }\text{ }173\text{ }=\text{ }0\Rightarrow 51x\text{ }+\text{ }15y\text{ }\text{ }50z\text{ }+\text{ }173\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ae593841670ed40e347399ad7d56e53a_l3.png "Rendered by QuickLaTeX.com")
Therefore,
the required equation of plane is:
![\[~51x\text{ }+\text{ }15y\text{ }\text{ }50z\text{ }+\text{ }173\text{ }=\text{ }0.\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0c2300d7041b0813ccb313394eb4dd32_l3.png "Rendered by QuickLaTeX.com")
