Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point $(2,3)$.

$A$ line passing through the point $P(1, 1)$ and parallel to the line $x - y = 5$ cuts the line $x + 3y - 2 = 0$ at $Q$. Then twice the length of the segment $PQ$ is

The $y$-intercept of the line passing through $A(6, 1)$ and perpendicular to the line $x - 2y = 4$ is:

If a straight line is at a distance of $10$ units from the origin and the perpendicular drawn from the origin to it makes an angle of $\frac{\pi}{4}$ with the negative $X$-axis in the negative direction,then the equation of that line is:

The equation of the line passing through $(1, -2)$ and perpendicular to the $y$-axis is:

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?