Chapter 9 Quadratic Relations and Conic Sections - 9.5 Graph and Write Equations of Hyperbolas - 9.5 Exercises - Problem Solving - Page 647: 43b

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The waves are travelling at the same speed. If $d_1$ is the distance between a focus and the closed portion of the hyperbola, and $d_2$ is the distance between a focus and the same section of the hyperbola, then $d_2$ - $d_1$ is constant. As you go around the hyperbola, $d_2$ and $d_1$ increase/decrease at the same pace. Because the waves' speeds are $d_2$ and $d_1$, the paths of their intersections will follow a hyperbola's path.