In what ratio does the point $(8, 4)$ divide the line segment joining the points $(5, -2)$ and $(9, 6)$?

The distance between the points $(a \cos \alpha, a \sin \alpha)$ and $(a \cos \beta, a \sin \beta)$ is

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:

If $C$ is the reflection of $A(2, 4)$ in the $x$-axis and $B$ is the reflection of $C$ in the $y$-axis, then $|AB|$ is

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

Find a point whose $x$-coordinate and $y$-coordinate are equal,and which is equidistant from the points $A(1, 0)$ and $B(0, 3)$.