A straight line passes through the point $P(2,√3)$ and makes an angle of $60°$ with X- axis. If this line intersects another line having equation $x+y√3 =12$ at $Q$, find the length of $PQ$.
My Attempt:
Here, the slope of the line passing through $P$ and making an angle of $60°$ with X- axis is: $$m=Tan60°$$ $$m=√3$$ Then,
Its equation is $$y-y_1=m(x-x_1)$$ $$y-√3 = √3 (x-2)$$ $$y-√3 = √3 x-2√3$$ $$√3 x - y - √3 =0$$
I could not continue from here. Please help to complete it.
You're doing well so far.
You have two lines now:
$\sqrt3 x - y - \sqrt3 =0$ and $x+y\sqrt3 =12$
You need to solve this pair of simultaneous equations to find $Q$
I would probably rearrange the first to make it $y=\sqrt3x-\sqrt3$
Then substitute that into the second:
$x+y\sqrt3 =12 \Rightarrow x+(\sqrt3x-\sqrt3)\sqrt3 =12$
$x+3x-3 =12$
and continue to find $x$ and $y$ etc.