The line segment joining $A(2, -7)$ and $B(6, 5)$ is divided into $4$ equal parts by the points $P, Q$ and $R$ such that $AP = PQ = QR = RB$. The midpoint of $PR$ is

Find the distance between the points $P(-2, 3)$ and $Q(4, -1)$.

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

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 coordinates of a point are $(0, 1)$ and the ordinate of another point is $-3$. If the distance between the two points is $5$,then the abscissa of the other point is:

Find the coordinates of the point which divides the line segment joining the points $(a + b, a - b)$ and $(a - b, a + b)$ externally in the ratio $a : b$.