Find the number of points on the $x$-axis that are at a distance of $c$ units from $(2, 3)$,where $c < 3$.

Three points $P, Q,$ and $R$ lie on the line segment joining points $A(-6, 8)$ and $B(8, -6)$ such that $AP = PQ = QR = RB$. Find the coordinates of $R$.

If the coordinates of two opposite vertices of a square are $(a, b)$ and $(b, a)$,find the area of the square.

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

The point which divides externally the line joining the points $(a + b, a - b)$ and $(a - b, a + b)$ in the ratio $a:b$ is

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