Find the equation of a straight line that makes an intercept of $3$ units on the negative direction of the $y$-axis and is inclined at an angle of ${\tan ^{ - 1}}\left( {\frac{3}{5}} \right)$ with the $x$-axis.

The polar equation of the line perpendicular to the line $\sin \theta - \cos \theta = \frac{1}{r}$ and passing through the point $\left(2, \frac{\pi}{6}\right)$ is

Find the equation of a straight line which passes through the point $(-1, -1)$ and makes an angle $150^{\circ}$ with the positive direction of the $X$-axis.

The equation of a straight line,where the length of the perpendicular from the origin is $4$ units and the line makes an angle of $120^{\circ}$ with the $x$-axis,is:

$A$ line passing through the point $A(-5, -4)$ intersects the lines $x + 3y + 2 = 0$,$2x + y + 4 = 0$,and $x - y - 5 = 0$ at points $B$,$C$,and $D$ respectively. If $\left( \frac{15}{AB} \right)^2 + \left( \frac{10}{AC} \right)^2 = \left( \frac{6}{AD} \right)^2$,find the equation of the line.

Find the value of $k$ for which the line $(k-3) x - (4-k^2) y + k^2 - 7k + 6 = 0$ passes through the origin.