$A$ 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

The area of the parallelogram formed by the lines $y = mx$,$y = mx + 1$,$y = nx$,and $y = nx + 1$ is equal to

Let $A \equiv (0, 1)$,$B \equiv (2, 0)$,and point $P$ be a point on the line $4x + 3y + 9 = 0$. Find the coordinates of point $P$ such that $|PA - PB|$ is maximized.

Among the statements:
$(S1)$ : If $A(5, -1)$ and $B(-2, 3)$ are two vertices of a triangle, whose orthocentre is $(0, 0)$, then its third vertex is $(-4, -7)$ and
$(S2)$ : If positive numbers $2a, b, c$ are three consecutive terms of an $A.P.$, then the lines $ax + by + c = 0$ are concurrent at $(2, -2)$.

The equation of one of the sides of an isosceles right-angled triangle,whose hypotenuse is $3x + 4y = 4$ and the vertex opposite to the hypotenuse is $(2, 2)$,is:

The number of integer values of $m$,for which the $x$-coordinate of the point of intersection of the lines $3x + 4y = 9$ and $y = mx + 1$ is also an integer,is