What is the total number of circles passing through the two fixed points?

Options

A. 1
B. 2
C. 4
D. Infinite

Answer

To solve the question of how many circles can pass through two fixed points, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Fixed Points**:
Let's denote the two fixed points as A and B. These points can be anywhere on a plane.

2. **Understanding Circle Properties**:
A circle is defined by its center and radius. For a circle to pass through two fixed points, the center of the circle must lie on the perpendicular bisector of the line segment connecting the two points A and B.

3. **Construct the Perpendicular Bisector**:
Draw the line segment AB connecting points A and B. The perpendicular bisector of this segment is the line that is perpendicular to AB and divides it into two equal halves at its midpoint.

4. **Determine Possible Circle Centers**:
Any point on the perpendicular bisector can serve as the center of a circle that passes through points A and B. Since the perpendicular bisector extends infinitely in both directions, there are infinitely many points along this line.

5. **Conclusion**:
Therefore, there are infinitely many circles that can pass through the two fixed points A and B.

### Final Answer:
The total number of circles passing through the two fixed points is **infinite**.