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Chapter 11: Problem 3

Give the property that defines all hyperbolas.

Short Answer

Expert verified
Answer: The defining property of a hyperbola is that it is the set of all points (x, y) in the coordinate plane such that the difference of the distances between the two fixed points called foci is constant. This property differentiates the hyperbola from other conic sections like ellipses, parabolas, and circles.

Step by step solution

01

Definition of a hyperbola

A hyperbola is a type of conic section that can be defined as the set of all points (x, y) in the coordinate plane such that the difference of the distances between the two fixed points called foci is constant. This property differentiates the hyperbola from the other conic sections.
02

Equation of a hyperbola

The standard equation of a hyperbola with the center at the origin (0, 0) is given by: \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] Here, a and b are positive constants, and they represent the distance from the center to the vertices and the distance from the center to the co-vertices respectively.
03

Foci of a hyperbola

The foci of the hyperbola are two fixed points, and they lie on the major axis (the axis that goes through the vertices) of the hyperbola. The distance between the center (0, 0) and each focus is given by the value c, which can be calculated using the following formula: \[c = \sqrt{a^2 + b^2}\] In summary, the property that defines all hyperbolas is the difference of the distances between any point (x, y) on the hyperbola and the two foci remaining constant.

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