교과서 질문
Finding Polar Areas
Find the areas of the regions in Exercises 9–18.
Shared by the circles r = 1 and r = 2 sin θ
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Ch. 11 - Parametric Equations and Polar Coordinates
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
11장, 문제 11.7.20
Hyperbolas and Eccentricity
In Exercises 17-24, find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.
y² − x² = 4
검증된 단계별 안내1
Rewrite the given hyperbola equation \(y^{2} - x^{2} = 4\) in the standard form by dividing both sides by 4, resulting in \(\frac{y^{2}}{4} - \frac{x^{2}}{4} = 1\).
Identify the values of \(a^{2}\) and \(b^{2}\) from the standard form. Here, \(a^{2} = 4\) and \(b^{2} = 4\). Since the \(y^{2}\) term is positive and comes first, the hyperbola opens vertically.
Calculate \(c^{2}\) using the relationship for hyperbolas: \(c^{2} = a^{2} + b^{2}\). Substitute the values to get \(c^{2} = 4 + 4\).
Find the eccentricity \(e\) using the formula \(e = \frac{c}{a}\). Take the square root of \(c^{2}\) to find \(c\), then divide by \(a\) (which is the square root of \(a^{2}\)).
Determine the coordinates of the foci, which lie along the transverse axis (vertical axis here) at \((0, \pm c)\), and write the equations of the directrices, which are horizontal lines given by \(y = \pm \frac{a^{2}}{c}\).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Standard Form of a Hyperbola
A hyperbola can be expressed in the standard form (y²/a²) - (x²/b²) = 1 or (x²/a²) - (y²/b²) = 1. This form helps identify the values of a² and b², which are essential for determining the hyperbola's shape, orientation, and key features like vertices and asymptotes.
추천 영상:
3:40
Circles in Standard Form Example 1
Eccentricity of a Hyperbola
Eccentricity (e) measures how 'stretched' a hyperbola is and is defined as e = c/a, where c² = a² + b². It is always greater than 1 for hyperbolas, indicating the distance of the foci from the center relative to the vertices.
추천 영상:
5:50Asymptotes of Hyperbolas
Foci and Directrices of a Hyperbola
The foci are two fixed points located along the transverse axis, at a distance c from the center. The directrices are lines perpendicular to the transverse axis, located at a distance a/e from the center. Both are used to define and graph the hyperbola accurately.
추천 영상:
5:22Foci and Vertices of Hyperbolas
관련 실천
교과서 질문
Cartesian to Polar Equations
Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.
(x + 2)² + (y − 5)² = 16"
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교과서 질문
Finding Cartesian from Parametric Equations
Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.
x=√(t+1), y=√t, t ≥ 0
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교과서 질문
Lines
Sketch the lines in Exercises 45–48 and find Cartesian equations for them.
r cos (θ + π/3) = 2
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교과서 질문
Hyperbolas and Eccentricity
Exercises 25–28 give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the xy-plane. In each case, find the hyperbola’s standard-form equation in Cartesian coordinates.
Eccentricity: 1.25
Foci: (0, ±5)
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교과서 질문
Polar Coordinates
Exercises 19–22 give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.
e = 1/3, r sin θ = −6
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