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

The distance of the midpoint of the line segment joining the points $(a \sin \theta, 0)$ and $(0, a \cos \theta)$ from the origin is:

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

The polar coordinates of the point whose Cartesian coordinates are $(-2, -2)$ are given by

Consider three points $P = (-\sin(\beta - \alpha), -\cos \beta)$,$Q = (\cos(\beta - \alpha), \sin \beta)$,and $R = (\cos(\beta - \alpha + \theta), \sin(\beta - \theta))$,where $0 < \alpha, \beta, \theta < \frac{\pi}{4}$. Then:

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