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Chapter 6: Problem 74

Explain how the central rectangle of a hyperbola can be used to sketch its asymptotes.

Short Answer

Expert verified
The central rectangle of a hyperbola is used to sketch its asymptotes by extending lines from the vertices or corners of the rectangle. These lines intersect at the center of the hyperbola, and cross over to form the asymptotes, which are the boundaries that the hyperbola will approach but never reach.

Step by step solution

01

Define a Hyperbola

A hyperbola is a type of conic section formed by intersecting a right circular cone by a plane. The hyperbola has two branches and these branches are symmetric about the line through their foci.
02

Central Rectangle in a Hyperbola

The central rectangle in a hyperbola is formed by the lines \(y-a = \pm (x-h)\) in a horizontal hyperbola and lines \(x-h = \pm (y-k)\) in a vertical hyperbola. The vertices of the central rectangle are the intersection points between the hyperbola and its conjugate.
03

Drawing Asymptotes Using the Central Rectangle

The asymptotes of a hyperbola are the diagonals of the central rectangle. They provide the boundary that the hyperbola will approach but never reach. To sketch the asymptotes, draw diagonal lines extending from each corner of the rectangle, thus bisecting its corners. These lines will intersect at the center of the hyperbola, and will extend to form the asymptotes.

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