Question

Find the value of $\theta$ and p, if the equation $xcos\theta +ysin\theta =p$ is the normal form of the line $\sqrt{3}x+y+2=0.$

Answer 1

$\text{The normal form of a line is}$

$xcos\theta +ysin\theta =p...\left(i\right)$

$\text{Let us try to write down the equation}$

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

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

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

$\Rightarrow -\frac{\sqrt{3}}{2}x-\frac{y}{2}=1$

$\left[\text{Dividing both sides by - 2}\right]$

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

$\text{Comparing equations (i) and (ii) we get,}$

$cos\theta =-\frac{\sqrt{3}}{2},andp=1$

$\Rightarrow \theta ={210}^{\circ }=\frac{7\pi }{6}andp=1$