Interaction between Two Lines

Given vertices $A(1, 1), B(4,-2) \& C(5, 5)$of a triangle, find the equation of the perpendicular dropped from $C$ to the interior bisector of the angle $A.$

The vertices of a triangle $∆OBC$ are $O \left(0, 0\right), B\left(-3, -1\right), C\left(-1, -3\right)$ find the equation of the line parallel to $BC$ and intersecting the sides $OB \& OC$, whose perpendicular distance from the point $\left(0, 0\right)$ is half.

The condition that the diagonals of the parallelogram formed by the lines ax + by + c = 0; ax + by + c' = 0; a'x + b'y + c = 0 & a'x + b'y + c' = 0 are at right angles is

Line $ax+by+p=0$ makes an angle $\frac{\mathrm{\pi }}{4}$ with $x\cos \alpha +y\sin \alpha =p, p\in R^{+}$. If these lines and the line $x\sin \alpha -ycos\alpha =0$ are concurrent, then

Equation of the line passing through $(1,2)$ and parallel to the line $y=3x-1$ is

For the straight lines $4x + 3y - 4 = 0$ and $5x + 12y + 6= 0$. find the equation of the bisector of the angle which contains the origin.

For the straight lines $4x + 3y - 3 = 0$ and $5x + 12y + 4 = 0$ find the equation of the bisector of the angle which contains the origin.

For the straight lines $4x + 3y - 2 = 0$ and $5x + 12y + 3 = 0$ find the equation of the bisector of the angle which contains the origin.

For the straight lines $4x + 3y - 1 = 0$ and $5x + 12y + 2 = 0$ find the equation of the bisector of the angle which contains the origin.

Find the equations of the bisectors of the angles between the straight lines $3x+4y+3=0$ and $3x-4y+5=0$

Find the equations of the bisectors of the angles between the straight lines $3x+4y+1=0$ and $3x-4y+3=0$

Find the equations of the bisectors of the angles between the straight lines $3x+4y+6=0$ and $3x-4y+2=0$

Find the equation of the obtuse angle bisector of lines $4x - 3y + 30 = 0$ and $8y-6x-10=0.$

Find the equation of the obtuse angle bisector of lines $4x - 3y + 20 = 0$ and $8y-6x-10=0.$

Find the equation of the obtuse angle bisector of lines $8x - 6y + 10 = 0$ and $12y- 9x +5 = 0.$

Find the equation of the obtuse angle bisector of lines $4x - 3y + 10 = 0$ and $6x - 8y + 5 = 0.$

For the straight lines $4x + 3y - 6 = 0$ and $5x + 12y + 9 = 0$ find the equation of the bisector of the angle which contains the origin.