Question

Find the value of θ and p, if the equation x cos θ + y sin θ = p is the normal form of the line \[\sqrt{3}x + y + 2 = 0\].

Answer 1

The normal form of a line is
x cos θ + y sin θ = p         ... (1)
Let us try to write down the equation

\[\sqrt{3}x + y + 2 = 0\] in its normal form.

\[\text { Now, } \sqrt{3}x + y + 2 = 0\]

\[ \Rightarrow \sqrt{3}x + y = - 2\]

\[ \Rightarrow - \frac{\sqrt{3}}{2}x - \frac{y}{2} = 1 \left[\text {  Dividing both sides by } - 2 \right]\]

\[ \Rightarrow \left( - \frac{\sqrt{3}}{2} \right)x + \left( - \frac{1}{2} \right)y = 1 . . . (2)\]

Comparing equations (1) and (2) we get,

\[\cos\theta = - \frac{\sqrt{3}}{2}, \text { and} p = 1\]

\[ \Rightarrow \theta = {210}^\circ = \frac{7\pi}{6}\text {  and } p = 1\]