The equations of the sides $AB$,$AC$,and $BC$ of a $\triangle ABC$ are respectively $x-3y=0$,$3x-y=0$,and $x+y+4=0$. If $P$ and $Q$ are points on the line $3x-y+k=0$ passing through $B$ such that $PB:BQ=1:1$,then $k=$

Let the coordinates of the two points $A$ and $B$ be $(1, 2)$ and $(7, 5)$ respectively. The line $AB$ is rotated through $45^{\circ}$ in the anti-clockwise direction about the point of trisection of $AB$ which is nearer to $B$. The equation of the line in the new position is:

$A$ line passes through $(x_{1}, y_{1})$ and $(h, k)$. If the slope of the line is $m$,show that $k - y_{1} = m(h - x_{1})$.

Find the equation of the line passing through the point $(2, -3)$ such that the sum of its intercepts on the axes is $-2$.

In what ratio does the line $y - x + 2 = 0$ divide the line segment joining the points $(3, -1)$ and $(8, 9)$?

The equation of the straight line passing through the point $(4, 3)$ and making intercepts on the coordinate axes whose sum is $-1$ is