Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbola
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 (y^2/a^2) - (x^2/b^2) = 1 for vertical hyperbolas, or (x^2/a^2) - (y^2/b^2) = 1 for horizontal hyperbolas. Understanding its structure is essential for graphing and analyzing its properties.
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Introduction to Hyperbolas
Foci of a Hyperbola
The foci of a hyperbola are two fixed points located along the transverse axis, which help define the shape of the hyperbola. For a hyperbola in the form (y^2/a^2) - (x^2/b^2) = 1, the distance from the center to each focus is given by c = √(a^2 + b^2). The foci play a crucial role in determining the hyperbola's eccentricity and its reflective properties.
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Foci and Vertices of Hyperbolas
Asymptotes of a Hyperbola
Asymptotes are lines that the branches of a hyperbola approach but never touch. For a hyperbola in the form (y^2/a^2) - (x^2/b^2) = 1, the equations of the asymptotes are given by y = ±(a/b)x. These lines provide a framework for sketching the hyperbola and indicate its direction and growth as the values of x or y become very large.
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