Which term best describes the set of all points in a plane for which the sum of the distances to two fixed points equals a certain constant?

The term that best describes the set of all points in a plane for which the sum of the distances to two fixed points equals a certain constant is called an ellipse.

Explanation

An ellipse is a geometric shape characterized by its focal points. In this context, the two fixed points are referred to as the foci of the ellipse. The defining property of an ellipse is that for any point on the ellipse, the sum of the distances from that point to the two foci is constant. This constant is greater than the distance between the two foci.

In other words, if we denote the two fixed points as $F_1$ and $F_2$, and let $d_1$ be the distance from any point $P$ on the ellipse to $F_1$ and $d_2$ the distance from $P$ to $F_2$, then the relationship can be expressed as:

$d_1 + d_2 = k$

where $k$ is a constant that satisfies the condition $k > |F_1F_2|$, where $|F_1F_2|$ is the distance between the two foci.

In conclusion, the concept of an ellipse arises in various applications in mathematics, physics, and engineering, making it a fundamental shape in geometry.