Question

A straight line makes an angle $\frac{\pi }{3}$ with positive direction of x-axis measured in anti clock direction and makes positive intercept on y-axis and the line is at a perpendicular distance of $5$ units from origin then its equation is?

• A. $x+\sqrt{3}y=10$
• B. $x=\sqrt{3}y+10$
• C. $y=\sqrt{3}x+10$
• D. $y=\sqrt{3}x-10$

Answer 1

Answer: (A)

$x+\sqrt{3}y=10$

Given, perpendicular distance of line from origin is $p=5$ units.

And angle makes with positive direction of $x$ is $\omega =\frac{\pi }{3}.$

Hence, equation of line which makes angle $\omega$ with positive direction of $x$

And

line has distance $p$ from the origin is $x\cos \omega +y\sin \omega =p$

$\therefore x\cos \frac{\pi }{3}+y\sin \frac{\pi }{3}=5$

$\frac{1}{2}x+\frac{\sqrt{3}}{2}y=5$

$x+\sqrt{3}y=10$