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The blue line is a hyperbola with foci at the red dots.
Reflection Property for Hyperbolas: any particle that is directed towards a focus of a hyperbola will be reflected towards the other focus.
This reflection property makes hyperbolas useful in telescopes, where a hyperbola shaped mirror can be used to direct light towards a point. There is a nice diagram of this in our text-note the reflecting telescope uses parabolas, ellipses, and hyperbolas!
Nice Animation (it may take awhile to load) that shows this in a different way. You may have to watch it a few times to understand it!