Find the equation of the line perpendicular to the line $x-7y+5=0$ and having $x$-intercept $3$.

$A$ 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.

$A$ line through the point $A(2,0)$ which makes an angle of $30^{\circ}$ with the positive direction of $x$-axis is rotated about $A$ in clockwise direction through an angle $15^{\circ}$. Then the equation of the straight line in the new position is

Find the equation of the line passing through the point $(0, 2)$ making an angle of $\frac{2 \pi}{3}$ with the positive $x$-axis. Also,find the equation of the line parallel to it and crossing the $y$-axis at a distance of $2$ units below the origin.

If a line $L$ passing through a point $A(2, 3)$ intersects another line $4x - 3y - 19 = 0$ at the point $B$ such that $AB = 4$,then the angle made by the line $L$ with the positive $X$-axis in the anti-clockwise direction is

Suppose a line makes an angle of $120^{\circ}$ with the positive direction of $X$-axis. If the length of the perpendicular from the origin to that line is $4$,then the equation of the line is