# Find the values of $\theta$ and $p$, if the equation $x\: \cos \theta + y \sin \theta = p$ is the normal form of the line $\sqrt 3 x+y+2=0.$

$\begin {array} {1 1} (A)\;\theta = \large\frac{5\pi}{6}\: and \: p = 1 & \quad (B)\;\theta = \large\frac{5\pi}{6}\: and \: p = -1 \\ (C)\;\theta = \large\frac{7\pi}{6}\: and \: p = 1 & \quad (D)\;\theta = \large\frac{7\pi}{6}\: and \: p = -1 \end {array}$

Toolbox:
• The normal form of a line is $x \cos \theta + y \sin \theta = p$-----------(1)
Equation of the given line is $\sqrt 3 x+y+2=0$
This equation can also be written as
$-\sqrt 3 x - y=2$ dividing on both sides by 2 we get, $-\large\frac{\sqrt 3}{2}$$x-\large\frac{1}{2}$$y=1$
comparing this equation with equation (1) we get,
$\cos \theta = -\large\frac{\sqrt 3}{2}$ and $\sin \theta = \large\frac{1}{2}$, and $p=1$