Locus of the Centre of a Circle Passing through the Origin and Cutting Off a Length 2b from the Line x=c

To find the locus of the center of a circle passing through the origin and cutting off a length 2b from the line x=c, we can follow these steps:

1. Understand the problem:
- We are given a circle that passes through the origin (0,0) and cuts off a length of 2b from the line x=c.
- We need to determine the locus (set of all possible points) of the center of this circle.

2. Define the problem:
- The locus of the center of the circle represents all the points that satisfy the given conditions.
- We need to find an equation that describes this locus.

3. Analyze the problem:
- Let's consider a point (h, k) as the center of the circle.
- The distance between the center and the origin is √(h^2 + k^2), which is the radius of the circle.
- The circle passes through the origin, so the radius is equal to √(h^2 + k^2).
- The line x=c intersects the circle at two points. The distance between these two points is 2b, which is twice the radius of the circle.

4. Find the equation of the circle:
- Since the circle passes through the origin, the equation of the circle can be written as:
x^2 + y^2 = (h^2 + k^2)

5. Determine the points of intersection:
- The equation of the line x=c can be written as:
x - c = 0
- Substituting this into the equation of the circle, we get:
(c - h)^2 + k^2 = (h^2 + k^2)
- Simplifying, we have:
c^2 - 2ch = 0
ch = c^2

6. Find the locus equation:
- Rearranging the equation, we get:
h = (c^2) / c
h = c

7. Conclusion:
- The locus of the center of the circle passing through the origin and cutting off a length 2b from the line x=c is given by the equation:
h = c, where h is the x-coordinate of the center and c is a constant value.

By following these steps, we have determined the locus equation of the center of the circle. This equation represents all the possible points that satisfy the given conditions.