If two vertical poles of heights $20\,m$ and $80\,m$ stand apart on a horizontal plane,then the height (in $m$) of the point of intersection of the lines joining the top of each pole to the foot of the other is:

$A$ line $L$ perpendicular to the line $5x - 12y + 6 = 0$ makes a positive intercept on the $Y$-axis. If the distance from the origin to the line $L$ is $2$ units and the angle made by the perpendicular drawn from the origin to the line $L$ with the positive $X$-axis is $\theta$,then $\tan \theta + \cot \theta =$

Points $A$ and $B$ are in the first quadrant; point $O$ is the origin. If the slope of $OA$ is $1$,the slope of $OB$ is $7$,and $OA = OB = r$,then the slope of $AB$ is:

Find the equation of the line passing through $(4, 6)$ and parallel to the line $3x - 7y + 2 = 0$.

$PS$ is the median of the triangle with vertices at $P(2, 2)$,$Q(6, -1)$,and $R(7, 3)$. The intercepts on the coordinate axes of the line passing through point $(1, -1)$ and parallel to $PS$ are respectively:

The equation of the hour hand at $4$ o'clock is