Question

# A straight line makes an angle $$\displaystyle\frac{\pi}{3}$$ 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+3y=10
B
x=3y+10
C
y=3x+10
D
y=3x10

Solution

## The correct option is B $$x+\sqrt{3}y=10$$ Given, perpendicular distance of line from origin is $$p=5$$ units. And angle makes with positive direction of $$x$$ is $$\omega =\dfrac{\pi }{3}.$$ Hence, equation of line which makes angle $$\omega$$ with positive direction of $$x$$And line has distance $$p$$ from the origin  is $$x\cos \omega +y\sin \omega =p$$ $$\therefore x\cos \dfrac{\pi }{3}+y\sin \dfrac{\pi }{3}=5$$ $$\dfrac{1}{2}x+\dfrac{\sqrt{3}}{2}y=5$$ $$x+\sqrt{3}y=10$$ Mathematics

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