Interaction between Two Straight Lines

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.

A straight line L is perpendicular to the line 5x - y = 1. The area of the triangle formed by the line L & the co-ordinate axes is 5. Find the equation of the line.

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

If the line $x+3y+2=0$ and its perpendicular line are conjugate with respect to $3x^2-5y^2=15$, then the equation to conjugate line is

The angle between the straight lines $3x+4y+9=0$ and $x-7\mathrm{y}-22=0$ is

The distance between the lines $3x+4y=9$ and $6x+8y=15$ is

The perpendicular bisector of line segment joining the points $P(1,4)$ and $Q(k,3)$ has $y$ intercept  $-4.$ Then a possible value of $k$ among the following is $\_\_\_$

Two vertices of a triangle are $\left(3,-2\right)$ and $\left(-2,3\right)$, and its orthocentre is $\left(-6,1\right)$. Then, the third vertex of this triangle cannot lie on the line

The equation of the straight line perpendicular to the straight line $3x-4y+7=0$ and passing through the point $(1, 1)$ is _____

The equation of the straight line parallel to the straight line $2x-3y+5=0$ and passing through the point $(2, 2)$ is ______

The equation of the straight line perpendicular to the straight line $3x+2=0$ and passing through the point $(2, -3)$ is _________

The angle included between the straight lines $2x-y+3=0$ and $x+2y+3=0$ is _________

The distance in between the straight lines $5x+3y=7$ and $15x+9y+14=0$ is ____

The distance between the lines $4x+3y=11$ and $8x+6y=15$ is _______

Find the joint equation of a pair of lines passing through the origin each of which making an angle of $30°$ with the line $3x+2y-11=0$.

The angle between the line joining the points $(1,-2),(3,2)$ and the line $x+2y-7=0$ is

The equation of the straight line which is perpendicular to the line $5x-2y=7$ and passing through the point of intersection of the lines $2x+3y-1=0$ and $3x+4y-6=0$ is

Given that lines $L_1:y=m_{a}x, L_2:y=m_{b}x$ and $L_3:y=m_{c}x$ make equal intercepts on the line $x+y=1$, then