Final answer:
A circle is a collection of points in a plane equidistant from a central point, thus the statement is false. An ellipse has two fixed points called foci, and the sum of the distances from the foci to any point on the ellipse is constant.
Step-by-step explanation:
The statement that a circle is the collection of points in a plane that are within a certain distance from a given point in the plane is false. Instead, a circle is a collection of points in a plane that are all exactly the same distance from a given point, which is the center of the circle. By definition, this distance is known as the radius.
In contrast, an ellipse is a closed curve in a plane where the sum of distances from any point on the ellipse to two fixed points (the foci) is constant. A circle can be seen as a special case of an ellipse where the two foci merge into a single point at the center, and therefore, the distance from this center point to any point on the circle is equal and consistent, forming the circle's radius.