If $(0, 0)$,$(-2, 1)$,and $(5, 2)$ are the vertices of a triangle,then the equation of the line passing through the centroid and parallel to the line $x - 2y = 6$ is:

$A$ 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 a straight line passing through $(-5, 6)$ and cutting off equal intercepts on the coordinate axes.

$A$ line passing through the point $P(a, 0)$ makes an acute angle $\alpha$ with the positive $x$-axis. Let this line be rotated about the point $P$ through an angle $\frac{\alpha}{2}$ in the clockwise direction. If in the new position,the slope of the line is $2-\sqrt{3}$ and its distance from the origin is $\frac{1}{\sqrt{2}}$,then the value of $3a^2 \tan^2 \alpha - 2\sqrt{3}$ is

$A$ straight line makes an angle of $135^\circ$ with the $x$-axis and cuts the $y$-axis at a distance of $-5$ from the origin. The equation of the line is

Reduce the following equation into slope-intercept form and find its slope and the $y$-intercept: $y=0$.