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Geometric Definition of Foci in Hyperbolic Navigation Calibration
Imagine you are a junior technician training at a maritime communications center. You are learning about older hyperbolic navigation systems like LORAN (Long Range Navigation). In these systems, a vessel's position is mapped along a hyperbolic curve, which is mathematically defined by its relationship to two fixed transmitting stations on land.
To demonstrate your understanding of the foundational geometry of these curves, write a brief essay explaining the role of these two transmitting stations, which represent the two fixed points of a hyperbola.
In your explanation, cover the following points:
- What is the mathematical term given to these two fixed points of a hyperbola?
- What is the defining geometric relationship between these two points and any point on the hyperbola? (Represent this relationship using mathematical notation, letting be the distance from a point on the hyperbola to the first transmitter and be the distance to the second transmitter.)
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Geometric Definition of Foci in Hyperbolic Navigation Calibration