Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.0 AMC 10/12 Find the answer

Given two fixed points $F_{1}(5,0)$ , $F_{2}(-5,0)$ , and a moving point $M$ satisfying $|MF_{1}|+|MF_{2}|=10$ , the trajectory of point $M$ is:

Pick one

Solution

If point $M$ and $F_{1}$ , $F_{2}$ can form a triangle, then $|MF_{1}|+|MF_{2}| > |F_{1}F_{2}|$ ,
since $|F_{1}F_{2}|=10$ , and the moving point $M$ satisfies $|MF_{1}|+|MF_{2}|=10$ ,
therefore, point $M$ is on the line segment $F_{1}F_{2}$ .
Hence, the correct choice is $\boxed{C}$ .
First, determine that point $M$ is on the line, then use the length relationship to determine that point $M$ is on the line segment $F_{1}F_{2}$ , thus reaching the conclusion.
This question examines the method of finding a trajectory, testing students' ability to analyze and solve problems, and is considered a medium-level question.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.