Question

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

• A. $\theta =\frac{3\pi }{2},p=-1$
• B. $\theta =\frac{7\pi }{6},p=1$
• C. $\theta =\frac{\pi }{2},p=1$
• D. None of these

Answer 1

The correct option is B $\theta =\frac{7\pi }{6},p=1$

We have,

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

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

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

This is same as $x\cos \theta +y\sin \theta =p$.

Therefore,

$\cos \theta =-\frac{\sqrt{3}}{2},\sin \theta =-\frac{1}{2}$ and $p=1$

$\theta =\frac{7\pi }{6}$ and $p=1$