Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.3 AMC 10/12 Find the answer

Let points $F_1(-3,0)$ and $F_2(3,0)$ . If the locus of point $M$ satisfying the condition $|MF_1|+|MF_2|=2m+1$ is an ellipse, and the locus of point $N$ satisfying the condition $\left| |NF_1|-|NF_2| \right| = 2m-1$ is a hyperbola, then the range of the real number $m$ is __________.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Analysis

This question examines the application of the concepts of ellipse and hyperbola, and it is a basic problem.

Solution

Given: $F_1(-3,0)$ , $F_2(3,0)$ , the distance between these two points is $|F_1F_2|=6$ .

Since the locus of point $M$ satisfying $|MF_1|+|MF_2|=2m+1$ is an ellipse,

then $2m+1 > 6$ ,

And since the locus of point $N$ satisfying $\left| |NF_1|-|NF_2| \right| = 2m-1$ is a hyperbola,

it follows that $2m-1 < 6$ ,

Therefore, $\dfrac{5}{2} < m < \dfrac{7}{2}$ .

Hence, the answer is $\boxed{\left( \dfrac{5}{2}, \dfrac{7}{2} \right)}$ .

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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.