The correct answer is option 'A' - Circular curve ranging.

Circular curve ranging is a method used in surveying to determine the coordinates and alignment of a circular curve on the ground. This method is commonly used in highway and railway alignment surveys, where curves are often encountered.

The two-theodolite method is a technique used to carry out circular curve ranging. It involves the use of two theodolites placed at known points on the ground, typically at the endpoints of the curve. The theodolites are set up and oriented to measure the angles between the tangent lines at the endpoints and the chord line connecting the endpoints.

The process of carrying out circular curve ranging using the two-theodolite method involves the following steps:

1. Set up the first theodolite at the starting point of the curve and the second theodolite at the endpoint of the curve.

2. Orient the theodolites so that they are level and pointing in the same direction.

3. Measure the angles between the tangent lines at the starting and ending points of the curve and the chord line connecting the two points.

4. Using trigonometry, calculate the radius, length, and coordinates of the circular curve based on the measured angles and the known distance between the two theodolites.

5. Mark the alignment of the circular curve on the ground using stakes or other markers.

The two-theodolite method is preferred for circular curve ranging because it provides accurate measurements of the angles and coordinates of the curve. It eliminates errors that may occur when using a single theodolite and allows for better control and adjustment of the curve alignment.

Other surveying methods such as tacheometric survey, geodetic survey, and astronomical survey are not specifically carried out using the two-theodolite method. These methods involve different techniques and instruments for measuring angles, distances, and coordinates in different surveying applications.