If the distance between the points $(a, 2)$ and $(3, 4)$ is $8$,then $a = $

Find the distance between the points $P \left( 3, -\frac{\pi}{6} \right)$ and $Q \left( 4, \frac{\pi}{3} \right)$ in polar coordinates.

The coordinates of the points $A, B, C$ are $(x_1, y_1)$,$(x_2, y_2)$,$(x_3, y_3)$ and $D$ divides the line $AB$ in the ratio $l : k$. If $P$ divides the line $DC$ in the ratio $m : k + l$,then the coordinates of $P$ are

The line joining $A(b \cos \alpha, b \sin \alpha)$ and $B(a \cos \beta, a \sin \beta),$ where $a \neq b,$ is produced to the point $M(x, y)$ such that $AM : MB = b : a$. Then,$x \cos \frac{\alpha+\beta}{2} + y \sin \frac{\alpha+\beta}{2}$ is equal to

If the line segment joining the points $A(a, b)$ and $B(c, d)$ subtends an angle $\theta$ at the origin,then $\cos \theta$ is equal to

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