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 $\sqrt{3}x+y-8=0 \quad\ldots\left(i\right)$
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 $cos\,30^{\circ}\left(x\right)+sin\,30^{\circ}\left(y\right)=4\quad\ldots\left(ii\right)$
Comparing $\left(ii\right)$ with $x\,cos\,\omega+y\,sin\,\omega=p$, we get $p=4$ and $\omega=30^{\circ}$