According to ques,:
The equation of the line perpendicular to:
![\[~x~-~3y\text{ }+\text{ }5\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-fc4fc59f12e7e0dac91950c9db1554ec_l3.png "Rendered by QuickLaTeX.com")
is
![\[3x\text{ }+\text{ }y\text{ }+~\lambda ~=\text{ }0,\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b0d848f864ec106920b8536ac0289363_l3.png "Rendered by QuickLaTeX.com")
Where, λ is a constant.
It passes through
![\[\left( 3,~-~2 \right).\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-476e748b6a30359970f546b17af0b077_l3.png "Rendered by QuickLaTeX.com")
Substitute the values in above equation, we get
![\[3\text{ }\left( 3 \right)\text{ }+\text{ }\left( -2 \right)\text{ }+\text{ }\lambda ~=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0f0b8e6d8ff7ec8ac8c1eb1583fdd4ec_l3.png "Rendered by QuickLaTeX.com")
![\[9\text{ }\text{ }2\text{ }+~\lambda ~=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5ccac80840cb984b91d601256b9996e4_l3.png "Rendered by QuickLaTeX.com")
Or ,
![\[\lambda ~=\text{ }\text{ }7\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-0c749835f8b10f63d22a573f12a6a61c_l3.png "Rendered by QuickLaTeX.com")
Now, substituting the value of
![\[\lambda ~=~-~7\text{ }in\text{ }3x\text{ }+\text{ }y\text{ }+~\lambda ~=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7d841fc6da9eefe6ea426fd780c30251_l3.png "Rendered by QuickLaTeX.com")
, we have
![\[3x\text{ }+\text{ }y\text{ }\text{ }7\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8d0ec3f5706e3a6d6d1049c0854556b5_l3.png "Rendered by QuickLaTeX.com")
∴ The required line is
![\[3x\text{ }+\text{ }y\text{ }\text{ }7\text{ }=\text{ }0.\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5a23436a0652787db08ae41888feb265_l3.png "Rendered by QuickLaTeX.com")