The length of the perpendicular from the orig | vedclass

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(A) The normal form of a line is $x \cos \alpha + y \sin \alpha = p$,where $p$ is the length of the perpendicular from the origin and $\alpha$ is the

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$x + \sqrt{3}y = 14$

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(A) The normal form of a line is $x \cos \alpha + y \sin \alpha = p$,where $p$ is the length of the perpendicular from the origin and $\alpha$ is the angle the normal makes with the positive $x$-axis.Given $p = 7$.The normal makes an angle of $150^{\circ}$ with the positive $y$-axis.The angle $\alpha$ with the positive $x$-axis is $150^{\circ} - 90^{\circ} = 60^{\circ}$.Substituting these values into the normal form equation:$x \cos 60^{\circ} + y \sin 60^{\circ} = 7$$x (1/2) + y (\sqrt{3}/2) = 7$$x + \sqrt{3}y = 14$

The length of the perpendicular from the origin to a line is $7$ and the line makes an angle of $150^{\circ}$ with the positive direction of the $y$-axis. Find the equation of the line.

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