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PP Section 11.5

# PP Section 11.5 - The Hyperbola Definition A hyperbola is...

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The Hyperbola

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Definition A hyperbola is the set of all points in a plane the difference of whose distances from two fixed points F 1 and F 2 (the foci) is a constant.
The derivation of the equation of a hyperbola is similar to the one done for an ellipse, working with the difference of the distances instead of the sum. When the foci are on the x -axis at (± c, 0) and the difference of distances is |PF 1 | – | PF 2 | = ±2 a, then the equation of the hyperbola is: 2 2 2 2 1 x y a b - = In this case, c 2 = a 2 + b 2

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2 2 2 2 1 x y a b - = Notice that: The x -intercepts are again ± a. The points ( a , 0) and (– a , 0) are the vertices of the hyperbola. x = 0 gives . Hence, no y -intercepts. y 2 = – b 2 2 2 2 2 1 1 x y a b = + Since , we get x 2 a 2 , so | x| ≥ a This means that the hyperbola consists of two parts, called its branches.
hyperbola. the of asymptotes the called are lines The x a b y ± = When sketching the graph of a hyperbola, it is useful to first draw its asymptotes, since the branches will approach them as | x | increases.

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