Short Answer 
The ellipse has a major axis of 8 units and a minor axis of 4 units, giving a semi-major axis of 4 units and a semi-minor axis of 2 units. As it moves, its center maintains a distance of 2 units to the x-axis and 4 units to the y-axis, creating a circular locus with a radius of 6 units centered at the point (4, 2).
Identify the Ellipse Properties
To determine the characteristics of the ellipse, first, note the specifications provided: the major axis is 8 units long and the minor axis is 4 units long. From these values, you can find the semi-major axis and semi-minor axis by dividing the lengths by 2: the semi-major axis measures 4 units, while the semi-minor axis measures 2 units.
Determine the Center’s Movement
As the ellipse slides between the coordinate axes, its center must remain equally distanced from both axes based on the lengths of the semi-axes. This means that the distances from the center of the ellipse will be the following: to the x-axis should always be 2 units (the semi-minor axis) and to the y-axis should always be 4 units (the semi-major axis). This constant relationship defines how the ellipse’s center moves across the coordinate plane.
Calculate the Locus of the Center
The path created by the center of the ellipse as it slides defines a circular locus. This circle has a radius equal to the sum of the semi-major and semi-minor axes, which is calculated as 4 + 2 = 6 units. Therefore, the locus is a circle centered at the point (4, 2) in the coordinate plane, with a radius of 6 units.