Find the slope of a line,which passes through the origin and the mid-point of the line segment joining the points $P(0, -4)$ and $B(8, 0)$.

If the line $2x - y - 4 = 0$ divides the line segment joining the points $(2, -1)$ and $(1, -4)$ at the point $(a, b)$ in the ratio $m:n$,then $4(a - b(\frac{m}{n})^2) = $

What is the square root of the sum of the squares of the lengths of the perpendiculars drawn from the point $(-3, 4)$ to both axes?

If the polar coordinates of a point are $(2, \pi /3)$,find its Cartesian coordinates.

Consider three points $P = (-\sin(\beta - \alpha), -\cos \beta)$,$Q = (\cos(\beta - \alpha), \sin \beta)$,and $R = (\cos(\beta - \alpha + \theta), \sin(\beta - \theta))$,where $0 < \alpha, \beta, \theta < \frac{\pi}{4}$. Then:

Find the coordinates of the point that divides the line segment joining the points $(-3, 2)$ and $(3, -4)$ internally in the ratio $3 : 2$.