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

If $C$ is the reflection of $A(2, 4)$ in the $x$-axis and $B$ is the reflection of $C$ in the $y$-axis, then $|AB|$ is

The points which trisect the line segment joining the points $(0, 0)$ and $(9, 12)$ are

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

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 segment joining the points $A(a, b)$ and $B(c, d)$ subtends an angle $\theta$ at the origin,then $\cos \theta$ is equal to