$\begin{array}{1 1}(A)\;p = 4\: units\: and \: \alpha = 120^{\circ} \\(B)\; p = 4\: units\: and \: \alpha = 60^{\circ} \\(C)\; p = 8\: units\: and \: \alpha = 120^{\circ} \\(D)\;p = 8\: units\: and \: \alpha = 60^{\circ} \end{array} $

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- Equation of a line in the normal form is $ x \cos \alpha + y \sin \alpha=p$ where p is the perpendicular distance from the origin and $ \alpha$ is the angle between the perpendicular and the positive $x$ - axis.

Given equation is $ x-\sqrt 3y=-8$

This can be written as

$ \qquad x-\sqrt 3y=-8$

or $ \quad -x+\sqrt 3y=8$

$ \sqrt{(-1)^2+(\sqrt 3)^2}=\sqrt 4 = 2$

dividing by 2 on both sides we get,

$\qquad -\large\frac{x}{2}$$+\large\frac{\sqrt 3}{2}$$y=4$

$ \Rightarrow \bigg( -\large\frac{1}{2}$$\bigg)(x)+ \bigg( \large\frac{\sqrt 3}{2}$$\bigg)(y)=4$

This implies that $\cos \alpha = -\large\frac{1}{2}$ and $\sin \alpha = \large\frac{\sqrt 3}{2}$.

That is the angle is in the second quadrant .

$ \therefore \alpha = 120^{\circ}$

$ \Rightarrow x \cos 120^{\circ}+y \sin 120^{\circ}=4$

which is in the normal form.

Comparing the above equation with the normal form of the equation we get, $x \cos \alpha + y \sin \alpha=p$

Hence $p=4$

That is the perpendicular distance from the origin to the line is 4 units.

The angle $ \alpha = 120^{\circ}$

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