Question

Find the lines through the point (0, 2) making angles $\frac{\pi }{3}and\frac{2\pi }{3}$ with the x-axis. Also, find the lines parallel to them cutting the y-axis at a distance of 2 units below the origin.

Answer 1

$\text{Equation of the line passing through}(x_1,y_1)\text{and making angle}\theta \text{with the x-axis is,}(y-y_1)=tan\theta (x-x_1)\text{For the first line :}(x_1,y_1)=(0,2),\theta =\frac{\pi }{3}(y-y_1)=tan\theta (x-x_1)(y-2)=\left(tan\frac{\pi }{3}\right)(x-0)y-2=\sqrt{3}x\sqrt{3}x-y+2=0\text{For the second line :}(x_1,y_1)=(0,2),\theta =\frac{2\pi }{3}(y-y_1)=tan\theta (x-x_1)(y-2)=\left(tan\frac{2\pi }{3}\right)(x-0)y-2=\sqrt{3}x\sqrt{3}x-y+2=0\text{The line parallel to}\sqrt{3}x-y+2=0\text{and cutting y-axis at a distance of 2 units below the origin.}y=\sqrt{3}x-2\sqrt{3}x-y+2=0\text{The line parallel to}\sqrt{3}x+y-2=0\text{and cutting y-axis at a distance of 2 units below the origin.}y=-\sqrt{3}x-2\sqrt{3}x+y+2=0$