Angle Bisectors

Find the equation of the line which bisects the obtuse angle between the lines $x-2y+4=0$ and $4x-3y+2=0$.

Two sides of a rhombus $ABCD$ are parallel to the lines $y=x+2$ and $y=7x+3$. If the diagonals of the rhombus intersect at the point $\left(1,2\right)$ and the vertex $A$ is on the $y$-axis and possible co-ordinates of $A$ are $\left(0,a\right)$ and $\left(0,b\right)$ then $4\left(a+b\right)$ is equal to

The line $3x+2y=24$ meets the $y$-axis at $A$ & the $x$-axis at $B$. The perpendicular bisector of $AB$ meets the line through $\left(0,-1\right)$ parallel to $x$-axis at $C$. If the area of the triangle $ABC$ is $\lambda$ sq. units, then value of $\lambda$ is equal to

A triangle is formed by the lines whose equations are $AB : x+y-5=0, BC : x+7y-7=0$ and $CA : 7x+y+14=0.$ Find the bisector of the interior angle at $B$ and the exterior angle at $C$. Determine the nature of the interior angle at $A$ and find the equation of the bisector.