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

$A$ man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4}$. Three stones $A, B$ and $C$ are placed at the points $(1,1), (2,2)$ and $(4,4)$ respectively. Which of these stones is/are on the path of the man?

Find the equation of the line passing through the point $(0, 2)$ making an angle of $\frac{2 \pi}{3}$ with the positive $x$-axis. Also,find the equation of the line parallel to it and crossing the $y$-axis at a distance of $2$ units below the origin.

Find the equation of a line that makes an intercept of $-3$ on the $y$-axis and makes an angle of ${\tan ^{ - 1}}\frac{3}{5}$ with the $x$-axis.

In an isosceles triangle $ABC$,the coordinates of the points $B$ and $C$ on the base $BC$ are respectively $(1, 2)$ and $(2, 1)$. If the equation of the line $AB$ is $y = 2x$,then the equation of the line $AC$ is

If the straight line $2x - 3y + 17 = 0$ is perpendicular to the line passing through the points $(7, 17)$ and $(15, \beta)$,then $\beta$ equals