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Chapter 10: Problem 32

Write the equation of a hyperbola from the given information. Graph the equation. Place the center of each hyperbola at the origin of the coordinate plane. (Distance from the center of a hyperbola to a focus) \(^{2}=96\) ; endpoints of the transverse axis are at \((-\sqrt{32}, 0)\) and \((\sqrt{32}, 0) .\)

Short Answer

Expert verified
The equation of the hyperbola is \(\frac{x^{2}}{32} - \frac{y^{2}}{64} = 1\).

Step by step solution

01

Understand the given information

The given information specifies that the center of the hyperbola is at the origin of the coordinate plane, the foci are at a distance of \(sqrt{96}\) from the center, and the endpoints of the transverse axis are at \(-sqrt{32}\) and \(sqrt{32}\). This suggests the hyperbola opens leftward and rightward.
02

Determine the values of a and c

A hyperbola is defined by the equation \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) when it opens leftward and rightward with its center at the origin. We are given that the distance from the center to each focus is \(sqrt{96}\), which is c. We are also given that the distance from the center to each endpoint of the transverse axis is \(sqrt{32}\), which is a.
03

Use the relationship between a, b, and c to find b

The relationship between a, b, and c in a hyperbola is \(c^{2} = a^{2} + b^{2}\). Use this relationship to find b. Substituting the given values, we get \(96 = 32 + b^{2}\) which simplifies to \(b^{2} = 64\). So, b = 卤 \(sqrt{64} = 卤8\).
04

Use a and b to write the equation of the hyperbola

With a = \(sqrt{32}\) and b = 8, plug these values into the formula to get the equation of a hyperbola, which is \(\frac{x^{2}}{32} - \frac{y^{2}}{64} = 1\).
05

Sketch the hyperbola using the given points and the equation

First plot the endpoints of the transverse axis at \(-sqrt{32}, 0\) and \(sqrt{32}, 0\). Plot the center at the origin (0, 0). Sketch the hyperbola so it opens right and left from the center, passing through the endpoints of the transverse axis.

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Key Concepts

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

Conic Sections
Conic sections are shapes created by the intersection of a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas, and have many applications in fields like astronomy, physics, and engineering. The classification depends on the angle of the plane relative to the angle of the cone:
  • If the plane cuts parallel to the base of the cone, the conic section is a circle.
  • If the plane cuts at an angle but does not coincide with the base, an ellipse forms.
  • When the plane is parallel to the side of the cone, it results in a parabola.
  • Lastly, if the plane cuts through both nappes, then a hyperbola is formed.
The hyperbola, in particular, consists of two disconnected curves called branches that mirror each other. It has properties such as foci and asymptotes that guide the shape's appearance on the coordinate plane.
Coordinate Plane
The coordinate plane is a two-dimensional surface where each point is defined by a pair of numerical coordinates. These coordinates specify the point's location relative to two perpendicular lines called axes. The horizontal axis is the x-axis, and the vertical axis is the y-axis.
  • The point where these two axes intersect is known as the origin, which has coordinates (0, 0).
  • The plane is divided into four quadrants.
  • A point's position is given as (x, y), where x is the horizontal distance from the origin and y is the vertical distance.
This plane is essential for graphing equations and inequalities, as well as for visualizing geometric shapes like hyperbolas. Each conic section, including hyperbolas, can be graphed on this plane to analyze their transformations and intersections.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. They form the backbone of equations that describe geometric shapes on a coordinate plane, such as those in conic sections. For instance, in a hyperbola, the standard form of its equation is \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]helping to determine the shape's orientation and size based on parameters a and b.
  • Variables represent unknowns or quantities that can vary.
  • Constants are fixed values, often represented by numerals.
  • To solve or graph these expressions, one must manipulate them to isolate variables, find intercepts, or calculate slopes and tangents.
Understanding algebraic expressions is pivotal for rewriting and simplifying equations necessary for graphing conic sections accurately.
Graphing Equations
Graphing equations involves plotting points on the coordinate plane to visualize the solutions of equations. This process includes interpreting algebraic equations to create a graph that represents all solutions of the equation.
  • To graph a hyperbola, begin by plotting the center of the graph, which in many cases coincides with the origin.
  • Next, identify crucial elements such as foci, vertices, and axes鈥攚hich inform the broad shape and orientation.
  • Plot these key points and sketch the asymptotes, which the hyperbola branches will approach but never touch.
  • Finally, draw the hyperbola by connecting these key features, observing symmetry around the center due to its geometric properties.
This visual representation helps in understanding the behavior and properties of the equation's solutions, providing a clear picture of spatial relationships that algebraic solutions alone might not reveal.

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Most popular questions from this chapter

What is the length of the minor axis of the graph of \(\frac{x^{2}}{100}+\frac{y^{2}}{64}=1 ?\) \(\begin{array}{llll}{\text { A. } 12} & {\text { B. } 2 \sqrt{41}} & {\text { C. } 16} & {\text { D. } 20}\end{array}\)

Expand each binomial. $$ (x-2)^{4} $$

Write an equation of an ellipse in standard form with center at the origin and with the given characteristics. focus \((0,3 \sqrt{2}),\) height 19

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Which conic section is represented by the equation \(x^{2}+y^{2}=6 x-14 y-9 ?\) F. circle G. ellipse H. parabola J. hyperbola
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