The equation of the line making an angle of $135^\circ$ with the positive $x$-axis and having a $y$-intercept equal to $-2$ is:

The $y$-intercept made by the line $x \cos \alpha + y \sin \alpha = a$ is:

The equation of a straight line,where the length of the perpendicular from the origin is $4$ units and the line makes an angle of $120^{\circ}$ with the $x$-axis,is:

What is the equation of the line passing through $(-1, 1)$ and perpendicular to the line $2x + 3y + 4 = 0$?

Find the equation of the line that bisects the line segment joining the points $(5, 3)$ and $(4, 4)$ and makes an angle of $45^{\circ}$ with the positive direction of the $x$-axis.

The line $MN$ whose equation is $x-y-2=0$ cuts the $X$-axis at $M$ and the coordinates of $N$ are $(4,2)$. The line $MN$ is rotated about $M$ through $45^{\circ}$ in the anticlockwise direction. What is the equation of the line $MN$ in the new position?