The length of the perpendicular from the orig | vedclass

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(D) Let the normal form of the line be $x \cos \alpha + y \sin \alpha = p$,where $p$ is the length of the perpendicular from the origin and $\alpha$ i

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(D) Let the normal form of the line be $x \cos \alpha + y \sin \alpha = p$,where $p$ is the length of the perpendicular from the origin and $\alpha$ is the angle the perpendicular makes with the positive $x$-axis.Given $p = 9$.The perpendicular makes an angle of $120^{\circ}$ with the positive $y$-axis.Since the angle between the positive $x$-axis and positive $y$-axis is $90^{\circ}$,the angle $\alpha$ with the positive $x$-axis is $\alpha = 120^{\circ} - 90^{\circ} = 30^{\circ}$.Substituting these values into the normal form equation:$x \cos 30^{\circ} + y \sin 30^{\circ} = 9$$x \left( \frac{\sqrt{3}}{2} \right) + y \left( \frac{1}{2} \right) = 9$$x\sqrt{3} + y = 18$.

The length of the perpendicular from the origin to a line is $9$ and the perpendicular makes an angle of $120^{\circ}$ with the positive direction of the $y$-axis. Find the equation of the line.

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