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.

$$\begin{gathered} \mathrm{Now},\sqrt{3}x+y+2=0 \\ \Rightarrow \sqrt{3}x+y=-2 \\ \Rightarrow -\frac{\sqrt{3}}{2}x-\frac{y}{2}=1\left[\mathrm{Dividing}\mathrm{both}\mathrm{sides}\mathrm{by}-2\right] \\ \Rightarrow \left(-\frac{\sqrt{3}}{2}\right)x+\left(-\frac{1}{2}\right)y=1...\left(2\right) \end{gathered}$$

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

$$\begin{gathered} \cos \theta =-\frac{\sqrt{3}}{2},\mathrm{and}p=1 \\ \Rightarrow \theta ={210}^{\circ }=\frac{7\mathrm{\pi }}{6}\mathrm{and}p=1 \end{gathered}$$