If the midpoint of the line segment joining the points $(5, a)$ and $(b, 7)$ is $(3, 5)$,find $(a, b)$.

The straight line $3x + y = 9$ divides the line segment joining the points $(1, 3)$ and $(2, 7)$ in the ratio

Two particles $P$ and $Q$ located at the points with coordinates $P(t, t^3-16t-3)$ and $Q(t+1, t^3-6t-6)$ are moving in a plane. The minimum distance between them in their motion is

If the line joining the points $A(b \cos \alpha, b \sin \alpha)$ and $B(a \cos \beta, a \sin \beta)$ is extended to the point $N(x, y)$ such that $AN: NB = b: a$,then

Find the coordinates of the point that divides the line segment joining the points $(3, -1)$ and $(3, 4)$ externally in the ratio $2 : 3$.

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