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Chapter 10: Problem 4

Each hyperbola has two ________ that intersect at the center of the hyperbola.

Short Answer

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asymptotes

Step by step solution

01

Understand components of a hyperbola

A hyperbola, which is one of the types of conic sections, consists of two parts or 'branches'. It has a center, vertices, a transverse axis, and two asymptotes. The important component in this exercise is the asymptotes.
02

Identify what intersects at the center

Among all these components, it is the two asymptotes that intersect at the center of the hyperbola. The asymptotes provide a boundary that the branches of the hyperbola approach but never cross.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Asymptotes
Asymptotes are crucial lines in the anatomy of a hyperbola. Think of them as invisible guides. They create a boundary, gently steering the paths of hyperbola's branches without ever touching or crossing them.
In a hyperbola, there are two asymptotes that play vital roles. They intersect at a point called the center of the hyperbola.
Ultimately, these asymptotes help in sketching the hyperbola and predicting how its branches behave. One might think of them as roads that tell the branches the path they should never want to cross. Without these guiding lines, a hyperbola just wouldn鈥檛 be the same.
Conic Sections
Conic sections are the curves obtained by slicing a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas.
Hyperbolas are particularly fascinating because they are formed when a plane intersects both nappes (the top and bottom portions) of a double cone.
This results in two separate curves, or branches. These branches are what we recognize as a hyperbola.
  • Circle: Intersection of a plane parallel to the base.
  • Ellipse: Intersection of a plane that cuts through a cone at an angle.
  • Parabola: Plane parallel to a generator of the cone.
  • Hyperbola: Plane intersects both nappes.
Understanding conic sections is foundational as they appear in many real-world applications. Whether in satellite dishes or planetary orbits, the principles remain the same.
Center of Hyperbola
The center of a hyperbola is the magic point where several important lines and features meet. It is where the two asymptotes intersect.
You could say the center acts like the hyperbola鈥檚 heart, marking the midpoint between its branches as well.
The center is essential when plotting the hyperbola, especially because it also determines the grid layout for other key components like vertices and the transverse axis. It remains constant, anchoring the symmetries that define the hyperbola.
Transverse Axis
The transverse axis is one of the principal axes in a hyperbola. Think of it as the main street that cuts through the center of the hyperbola.
It stretches right across from one vertex to the other, essentially connecting the two main points where the hyperbola's branches turn closest to each other.
The length of this axis can vary and helps determine the hyperbola's shape. Understand that the transverse axis is perpendicular to the conjugate axis, which runs the opposite way through the hyperbola's center. The transverse axis plays a big role in describing the hyperbola's openness or closeness, directly affecting its eccentricity.

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Most popular questions from this chapter

GRAPHICAL REASONING Use a graphing utility to graph the polar equation \(r = 6[1+\cos(\theta - \phi)]\) for (a) \(\phi = 0\), (b) \(\phi = \pi/4\), and (c) \(\phi = \pi/2\). Use the graphs to describe the effect of the angle \(\phi\). Write the equation as a function of \(\sin\ \theta\) for part (c).

Consider the equation \(r=3\ \sin\ k\theta\). (a) Use a graphing utility to graph the equation for \(k=1.5\). Find the interval for \(\theta\) over which the graph is traced only once. (b) Use a graphing utility to graph the equation for \(k=1.5\). Find the interval for \(\theta\) over which the graph is traced only once. (c) Is it possible to find an interval for \(\theta\) over which the graph is traced only once for any rational number \(k\)? Explain.

In Exercises 39-54, find a polar equation of the conic with its focus at the pole. \(\textit{Conic}\) Parabola \(\textit{Eccentricity}\) \(e=1\) \(\textit{Directrix}\) \(y=-4\)

In Exercises 59-64, use a graphing utility to graph the polar equation. Find an interval for \(\theta\) for which the graph is traced only once. \(r=3\ -\ 8\ \cos\ \theta\)

In Exercises 23-48, sketch the graph of the polar equation using symmetry, zeros, maximum \(r\)-values, and any other additional points. \(r=-7\)
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