Ch. 7 - Conic Sections

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 7 - Conic Sections

Problem 15

Chapter 8, Problem 15

In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x²/100−y²/64=1

Verified step by step guidance

Identify the standard form of the hyperbola equation:

\frac{x^{2}}{100} - \frac{y^{2}}{64} = 1

. Since the x-term is positive and comes first, this is a hyperbola that opens left and right.

Determine the values of

a^2

and

b^2

from the denominators:

a^2 = 100

and

b^2 = 64

. Then find

a = \, \(\text{sqrt}\)(100) = 10

and

b = \, \(\text{sqrt}\)(64) = 8

Locate the vertices on the x-axis at

(��	extpm a, 0) = (	extpm 10, 0)

because the hyperbola opens horizontally.

Find the foci using the formula

c^2 = a^2 + b^2

. Calculate

c = \, \(\text{sqrt}\)(100 + 64) = \(\text{sqrt}\)(164)

. The foci are at

(	extpm c, 0)

Write the equations of the asymptotes using the slopes

extpm rac{b}{a} = 	extpm rac{8}{10} = 	extpm rac{4}{5}

. The asymptotes pass through the center (0,0), so their equations are

y = 	extpm rac{4}{5}x

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Form of a Hyperbola

A hyperbola in standard form is expressed as (x^2/a^2) - (y^2/b^2) = 1 or (y^2/a^2) - (x^2/b^2) = 1. This form helps identify the orientation of the hyperbola (horizontal or vertical), the vertices located at ±a along the transverse axis, and the relationship between a and b which determines the shape.

Foci of a Hyperbola

The foci are two fixed points located along the transverse axis of the hyperbola, found using c^2 = a^2 + b^2. They are essential in defining the hyperbola as the set of points where the difference of distances to the foci is constant. Knowing the foci helps in graphing and understanding the hyperbola's geometry.

Equations of the Asymptotes

Asymptotes are lines that the hyperbola approaches but never touches. For a hyperbola centered at the origin with horizontal transverse axis, the asymptotes have equations y = ±(b/a)x. These lines guide the shape and direction of the hyperbola branches and are crucial for accurate graphing.

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Introduction to Asymptotes

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Graph each ellipse and locate the foci. 25x²+4y² = 100

In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x2/100−y2/64=1

Verified video answer for a similar problem:

Key Concepts

Standard Form of a Hyperbola

Foci of a Hyperbola

Equations of the Asymptotes

In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x²/100−y²/64=1