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

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

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

What are the coordinates of the point that divides the line segment joining the points $(-3, -4)$ and $(-8, 7)$ externally in the ratio $7 : 5$?

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?

The polar coordinate of a point $P$ is $\left(2, -\frac{\pi}{4}\right)$. The polar coordinate of the point $Q$, which is such that the line joining $PQ$ is bisected perpendicularly by the initial line, is