Two satellites revolve around a planet in coplanar circular orbits in anticlockwise direction. Their period of revolutions are \(1\, hour\) and \(8\, hours\) respectively. The radius of the orbit of nearer satellite is \(2 \times 10^{3}\, {km}\). The angular speed of the farther satellite as observed from the nearer satellite at the instant when both the satellites are closest is \(\frac{\pi}{{x}}\, {rad} \,{h}^{-1}\) where \({x}\) is ..... .

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.