Given Information
- A circle cuts a chord of length 4a on the x-axis.
- The circle passes through a point on the y-axis, which is at a distance of 2b from the origin.

To Find
The locus of the center of this circle.

Solution

Step 1: Drawing the Diagram
Let's start by drawing the diagram to better understand the given information. We have a circle that cuts a chord of length 4a on the x-axis. The circle also passes through a point on the y-axis, which is at a distance of 2b from the origin.

[Diagram Explanation: Draw a coordinate system with x and y axes. Mark the origin as O(0,0). Draw a point A on the y-axis, which is at a distance of 2b from the origin. Draw a chord AB of length 4a on the x-axis. Join the points A and B with the center of the circle, C. Draw the circle with center C passing through points A and B.]

Step 2: Analyzing the Diagram
From the diagram, we can observe the following:
- The center of the circle lies on the perpendicular bisector of the chord AB.
- The center of the circle also lies on the y-axis, as it passes through point A.
- The distance of the center of the circle from the y-axis is 2b.

Step 3: Finding the Locus
Since the center of the circle lies on the perpendicular bisector of AB, we can find the equation of the perpendicular bisector using the midpoint formula.

The midpoint of AB is the point (2a, 0) as AB is of length 4a on the x-axis.

The slope of AB is 0 as it is parallel to the x-axis. Therefore, the slope of the perpendicular bisector is undefined.

Using the equation of a line with an undefined slope passing through the point (2a, 0), we get the equation of the perpendicular bisector as x = 2a.

Since the center of the circle lies on the y-axis at a distance of 2b from the origin, the equation of the locus of the center is x = 2a, y = ±2b.

Step 4: Simplifying the Locus
We can simplify the locus equation further by eliminating x.

Since x = 2a, substituting this value in the equation y = ±2b, we get y = ±b.

Therefore, the simplified equation of the locus of the center is y = ±b, which represents two parallel lines on the y-axis.

Hence, the correct answer is option 'B' - A parabola.