The correct answer is option 'D', which states that all the given options are correct.

Contact ratio in gear systems refers to the number of teeth in contact at any given moment. It is an important parameter that affects the smoothness and efficiency of gear operation. Let's discuss each statement individually to understand why they are true.

**a) Varies directly to the length of the arc of contact:**

The arc of contact is the portion of the gear tooth profiles that are in contact with each other during meshing. The longer the arc of contact, the more teeth are in contact at any given moment. Therefore, the contact ratio is directly proportional to the length of the arc of contact. This means that as the length of the arc of contact increases, the contact ratio also increases.

**b) Inversely proportional to the module:**

The module is a gear parameter that represents the size of the gear teeth. It is defined as the ratio of the pitch diameter to the number of teeth. As the module increases, the gear teeth become larger. When the gear teeth are larger, the number of teeth in contact at any given moment decreases. Hence, the contact ratio is inversely proportional to the module. This means that as the module increases, the contact ratio decreases.

**c) Inversely proportional to the circular pitch:**

Circular pitch is the distance between corresponding points on adjacent gear teeth measured along the pitch circle. It is determined by dividing the circumference of the pitch circle by the number of teeth. As the circular pitch increases, the distance between adjacent teeth increases. This leads to a decrease in the number of teeth in contact at any given moment. Therefore, the contact ratio is inversely proportional to the circular pitch. This means that as the circular pitch increases, the contact ratio decreases.

**d) All options are correct:**

From the explanations provided above, it is clear that all the given options are correct. The contact ratio varies directly with the length of the arc of contact, it is inversely proportional to the module and the circular pitch. Therefore, option 'D' is the correct answer.