According to ques,:
The equation is perpendicular to:
![\[~\surd 3x\text{ }\text{ }y\text{ }+\text{ }5\text{ }=\text{ }0~\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-30d1069950c7dbdc4f827fe80882d20d_l3.png "Rendered by QuickLaTeX.com")
equation and cuts off an intercept of 4 units with the negative direction of y-axis.
The line perpendicular to:
![\[~\surd 3x\text{ }\text{ }y\text{ }+\text{ }5\text{ }=\text{ }0~is\text{ }x\text{ }+\text{ }\surd 3y\text{ }+\text{ }\lambda \text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-20d8cf6c551a0892bf3b8671c75b0fc7_l3.png "Rendered by QuickLaTeX.com")
It is given that the line,
![\[~x\text{ }+\text{ }\surd 3y\text{ }+\text{ }\lambda \text{ }=\text{ }0~\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bc10e2602cf781032ed96d6505bde219_l3.png "Rendered by QuickLaTeX.com")
cuts off an intercept of 4 units with the negative direction of the y-axis.
This means that the line passes through (0,-4).
So,
Let us substitute the values in the equation
![\[~x\text{ }+\text{ }\surd 3y\text{ }+\text{ }\lambda \text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-6dbb807b223374dbe767de53b203ddc5_l3.png "Rendered by QuickLaTeX.com")
, we have
![\[0\text{ }\text{ }\surd 3\text{ }\left( 4 \right)\text{ }+\text{ }\lambda \text{ }=\text{ }0~\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5ad9318eb9f04e9268e03e18785e95c4_l3.png "Rendered by QuickLaTeX.com")
![\[\lambda \text{ }=\text{ }4\surd 3\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-d0b84f3953b81d96f7217f0b8a00b2df_l3.png "Rendered by QuickLaTeX.com")
Now, substitute the value of λ back, we get
![\[x\text{ }+\text{ }\surd 3y\text{ }+\text{ }4\surd 3\text{ }=\text{ }0\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-78a73ae2cca66c3a11a70b659a1f6232_l3.png "Rendered by QuickLaTeX.com")
∴ The required equation of line is
![\[x\text{ }+\text{ }\surd 3y\text{ }+\text{ }4\surd 3\text{ }=\text{ }0.\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-49a7680b27468b860b43c3a3a608b23a_l3.png "Rendered by QuickLaTeX.com")
