To find the relation between $G$ and $K,$ we start by analyzing the given equations.$\\$Kepler's third law states:$\\ T^2 = K r^3 \\$According to Newton's law of gravitation, the gravitational force $F$ is given by:$\\ F = \frac{G M m}{r^2} \\$For a planet in circular orbit, the gravitational force provides the necessary centripetal force:$\\ F = \frac{m v^2}{r} \\$Equating the gravitational force to the centripetal force, we have:$\\ \frac{G M m}{r^2} = \frac{m v^2}{r} \\$Simplifying, we get:$\\ G M = v^2 r \\$The orbital velocity $v$ can be expressed in terms of the period $T$ as:$\\ v = \frac{2 \pi r}{T} \\$Substitute $v$ in the equation $G M = v^2 r: \\ G M = \left(\frac{2 \pi r}{T}\right)^2 r \\$Simplifying, we have:$\\ G M = \frac{4 \pi^2 r^3}{T^2} \\$From Kepler's third law, $T^2 = K r^3.$ Substitute this into the equation:$\\ G M = \frac{4 \pi^2 r^3}{K r^3} \\$Cancel $r^3$ from both sides:$\\ G M = \frac{4 \pi^2}{K} \\$Rearranging gives the relation between $G$ and $K: \\ G K = \frac{4 \pi^2}{M} \\$Since the problem asks for the relation between $G$ and $K$ without considering $M,$ we assume $M$ is constant.$\\$Thus, the correct relation is:$\\ G K = 4 \pi^2 \\$Therefore, the correct option is Option 4.