$A$ straight line $x-2y-4=0$ is shifted parallel to it by $3$ units away from the origin and then rotated by an angle of $30^{\circ}$ in the anti-clockwise direction. If the slope of the new line formed is $m$,then the integral part of $m$ is

$L_1 \equiv ax-3y+5=0$ and $L_2 \equiv 4x-6y+8=0$ are two parallel lines. If $p, q$ are the intercepts made by $L_1=0$ and $m, n$ are the intercepts made by $L_2=0$ on the $X$ and $Y$ coordinate axes respectively,then the equation of the line passing through the points $(p, q)$ and $(m, n)$ is

If a line passing through the point $P(1, 1)$ makes an angle $\theta$ with the positive direction of the $x$-axis and intersects the coordinate axes at $A$ and $B$ such that $AB = OP$,where $O$ is the origin,then what is the value of $\tan\theta + \cot\theta$?

The equation of the straight line in the normal form which is parallel to the lines $x+2y+3=0$ and $x+2y+8=0$ and divides the distance between these two lines in the ratio $1:2$ internally is

The equation of the line passing through the point of intersection of the lines $2x + y - 4 = 0$ and $x - 3y + 5 = 0$ and lying at a distance of $\sqrt{5}$ units from the origin is:

Let a variable line of slope $m > 0$ passing through the point $(4, -9)$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is