The correct answer is option 'C': Analytical equations.

Coordinate geometry is a branch of mathematics that deals with the study of geometric shapes using algebraic methods. It combines concepts from algebra and geometry to analyze and describe the properties and relationships of points, lines, and shapes on a coordinate plane.

The distance formula, section formula, midpoint formula, and other calculations in coordinate geometry are all based on analytical equations. These equations involve the use of coordinates and algebraic expressions to derive the desired results.

- Distance Formula:
The distance formula is used to find the distance between two points in a coordinate plane. It is derived using the Pythagorean theorem and involves the square root of the sum of the squares of the differences between the x-coordinates and y-coordinates of the two points. The formula is:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

- Section Formula:
The section formula is used to find the coordinates of a point that divides a line segment into a given ratio. It is derived using the concept of linear interpolation and involves the use of the coordinates of the endpoints of the line segment and the ratio in which it is divided. The formula for finding the coordinates of the point (x, y) is:

x = (m1x2 + m2x1) / (m1 + m2)
y = (m1y2 + m2y1) / (m1 + m2)

- Midpoint Formula:
The midpoint formula is used to find the coordinates of the midpoint of a line segment. It is derived by taking the average of the x-coordinates and y-coordinates of the two endpoints of the line segment. The formula for finding the coordinates of the midpoint (x, y) is:

x = (x1 + x2) / 2
y = (y1 + y2) / 2

All of these formulas and calculations are based on analytical equations, which involve the use of coordinates, algebraic expressions, and mathematical operations. They allow us to perform geometric calculations and solve problems in a coordinate plane efficiently.

In conclusion, the distance formula, section formula, midpoint formula, and other calculations in coordinate geometry are all based on analytical equations. These equations involve the use of coordinates and algebraic expressions to derive the desired results.