Ch. 7 - Conic Sections

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Blitzer 8th Edition

Ch. 7 - Conic Sections

문제 15

8장, 문제 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

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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

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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