Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbola Definition
A hyperbola is a type of conic section formed by the intersection of a plane and a double cone. It consists of two separate curves called branches, which are mirror images of each other. The standard form of a hyperbola can be expressed as (x-h)²/a² - (y-k)²/b² = 1 for horizontal hyperbolas, or (y-k)²/a² - (x-h)²/b² = 1 for vertical hyperbolas, where (h, k) is the center.
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Asymptotes of a Hyperbola
Asymptotes are lines that the branches of a hyperbola approach but never touch. For a hyperbola in standard form, the equations of the asymptotes can be derived from the center and the values of 'a' and 'b'. Specifically, for a hyperbola centered at (h, k), the asymptotes are given by y - k = ±(b/a)(x - h), which helps in sketching the graph accurately.
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Foci of a Hyperbola
The foci of a hyperbola are two fixed points located along the transverse axis, which is the line segment that connects the vertices of the hyperbola. The distance from the center to each focus is denoted by 'c', where c² = a² + b². The foci play a crucial role in defining the shape of the hyperbola and are essential for understanding its geometric properties.
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