Question

Reduce the equation $\sqrt{3}x+y-8=0$ into normal form. Find the length of perpendicular from the origin to the given line and angle made by the perpendicular with $x$-axis.

• A. $8$, $120^{\circ }$
• B. $6$, $120^{\circ }$
• C. $4$, $30^{\circ }$
• D. $5,15^{\circ }$

Answer 1

Answer: (C)

Given equation is $\sqrt{3}x+y-8=0\dots \left(i\right)$
Dividing $\left(i\right)$ by $\sqrt{{\left(\sqrt{3}\right)}^2+{\left(1\right)}^2}=2$, we get
$\frac{\sqrt{3}}{2}x+\frac{1}{2}y=4$ or $cos30^{\circ }\left(x\right)+sin30^{\circ }\left(y\right)=4\dots \left(ii\right)$
Comparing $\left(ii\right)$ with $xcos\omega +ysin\omega =p$, we get $p=4$ and $\omega =30^{\circ }$