Find the equation of a line that makes an angle of $60^{\circ}$ with the $x$-axis and makes an intercept of $5$ units on the negative direction of the $y$-axis.

$A$ 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 ratio in which the line joining points $A(-1, -1)$ and $B(2, 1)$ divides the line segment joining points $C(3, 4)$ and $D(1, 2)$.

The equations of the diagonals of the square formed by the lines $x = 0,$ $y = 0,$ $x = 1,$ and $y = 1$ are

Find the equation of the line passing through the point $(2,2)$ and cutting off intercepts on the axes whose sum is $9$.

The equation of the straight line passing through the point $(3, 2)$ and perpendicular to the line $y = x$ is