Answer

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**Hint:**To solve this question, we need to relate the angular velocity of the satellite with that of the earth. Then by using the formula of the angular velocity in terms of the time period, we can relate the period of rotation of a relay satellite and that of the earth.

**Formula used:**The formula used to solve this question is given by

$\omega = \dfrac{{2\pi }}{T}$, here $\omega $ is the angular velocity of a rotating body whose period of rotation is equal to $T$.

**Complete step by step solution:**

A relay satellite is used for the communication between one place of the earth to the other place. It uses the radio waves which are sent as signals and are received by the antennas located on the earth’s surface. The term “relay” refers to the relay race, where the participating runners pass the baton to the next runner and reach the next leg of the race. Likewise, the signal is transmitted by the relay satellite between the space and the ground. These satellites keep on relaying the signal to each other until the signal finally reaches the ground station.

Now, the relay satellite is a geostationary satellite. We know that a geostationary satellite is a satellite which rotates with an angular velocity equal to the angular velocity of the earth. We know that the angular velocity is related with the time period of rotation by the formula

$\omega = \dfrac{{2\pi }}{T}$

So this means that if the angular velocities are equal, then the time periods of motion must also be equal.

Thus, the time period of a relay satellite is equal to the period of rotation of the earth about its axis.

**Hence, the correct answer is option C.**

**Note:**

Not only television programmes, but many types of information are communicated with the help of a relay satellite. For example, in weather forecasting and important defence and intelligence applications, relay satellites are employed.

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