The points $A (1, 3)$ and $C (5, 1)$ are the opposite vertices of a rectangle. The equation of the line passing through the other two vertices and having a gradient of $2$ is:

$A$ straight line passing through the point $(2, 2)$ intersects the lines $\sqrt{3}x + y = 0$ and $\sqrt{3}x - y = 0$ at points $A$ and $B$ respectively. Find the equation of the line $AB$ such that the triangle $OAB$ is an equilateral triangle,where $O$ is the origin.

Find the equation of a straight line which passes through the point $(-1, -1)$ and makes an angle $150^{\circ}$ with the positive direction of the $X$-axis.

The equations of the sides $AB$,$AC$,and $BC$ of a $\triangle ABC$ are respectively $x-3y=0$,$3x-y=0$,and $x+y+4=0$. If $P$ and $Q$ are points on the line $3x-y+k=0$ passing through $B$ such that $PB:BQ=1:1$,then $k=$

If the $x-$intercept of some line $L$ is double that of the line $3x + 4y = 12$ and the $y-$intercept of $L$ is half that of the same line,then the slope of $L$ is

Find the slope of the line,which makes an angle of $30^{\circ}$ with the positive direction of the $y$-axis measured anticlockwise.