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:

Find the equations of the sides $QR$ and $RP$ of a triangle $PQR$ where $P = (2, 1)$,and the sides $QR$ and $RP$ have slopes $m_1 = \frac{2}{\sqrt{3}}$ and $m_2 = -\frac{2}{\sqrt{3}}$ respectively,passing through the origin $(0, 0)$ for $QR$ and intersecting at $P(2, 1)$ for $RP$.

The equation of the line perpendicular to the line $\frac{x}{a} - \frac{y}{b} = 1$ and passing through the point at which it cuts the $x$-axis,is

The equation of the line,passing through the point $(a \cos^3 \theta, a \sin^3 \theta)$ and perpendicular to the line $x \cos \theta - y \sin \theta = a$,is

Write the equation of the lines for which $\tan \theta = \frac{1}{2}$,where $\theta$ is the inclination of the line and $x$-intercept is $4$.

$A$ straight line passing through the point $(2, 2)$ intersects the lines $\sqrt{3}x + y = 0$ and $\sqrt{3}x - y = 0$ at points $A$ and $B$ respectively. Find the equation of the line $AB$ such that the triangle $OAB$ is an equilateral triangle,where $O$ is the origin.