The length of the perpendicular from the origin to a line is $4$ units,and the perpendicular makes an angle of $30^{\circ}$ with the positive direction of the $x$-axis. Find the equation of the line.

The equation of the line passing through $(c, d)$ and parallel to $ax + by + c = 0$ is

Find the equation of the line passing through the point $(1, 2)$ and perpendicular to the line $y = 3x - 1$.

Find the slope of the line making an inclination of $60^{\circ}$ with the positive direction of the $x$-axis.

If each of the points $(a, 4)$ and $(-2, b)$ lies on the line joining the points $(2, -1)$ and $(5, -3)$,then the point $(a, b)$ lies on the line:

Let $u = \hat{i} - 2\hat{j}$ and $v = -3\hat{i} + 5\hat{j}$. Consider three points $P, Q,$ and $R$ having position vectors $-\frac{1}{7}\hat{i}$,$-\frac{1}{4}\hat{j}$,and $-2\hat{i} + 3\hat{j}$ respectively. Among these,the points on the line segment passing through $u$ and $v$ are