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Exercícios de cônicas · Physics Playground
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31 problemas com soluções resolvidas, de Algebra and Trigonometry.
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Algebra and Trigonometry
12.1 The Ellipse
Exemplo 1
Writing the Equation of an Ellipse Centered at the Origin in Standard Form
What is the standard form equation of the ellipse that has vertices ( ±8,0) and foci ( ±5,0)?
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Exemplo 2
Writing the Equation of an Ellipse Centered at a Point Other Than the Origin
What is the standard form equation of the ellipse that has vertices ( −2,−8) and ( −2,2) and foci ( −2,−7) and ( −2,1)?
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Exemplo 3
Graphing an Ellipse Centered at the Origin
Graph the ellipse given by the equation, (x^2)/9 +(y^2)/(25) =1. Identify and label the center, vertices, co-vertices, and foci.
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Exemplo 4
Graphing an Ellipse Centered at the Origin from an Equation Not in Standard Form
Graph the ellipse given by the equation 4x^2 +25y^2 =100. Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.
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Exemplo 5
Graphing an Ellipse Centered at (h, k)
Graph the ellipse given by the equation, (( x+2) ^2)/4 +(( y−5) ^2)/9 =1. Identify and label the center, vertices, co-vertices, and foci.
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Exemplo 6
Graphing an Ellipse Centered at (h, k) by First Writing It in Standard Form
Graph the ellipse given by the equation 4x^2 +9y^2 −40x+36y+100=0. Identify and label the center, vertices, co-vertices, and foci.
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Exemplo 7
Locating the Foci of a Whispering Chamber
The Statuary Hall in the Capitol Building in Washington, D.C. is a whispering chamber. Its dimensions are 46 feet wide by 96 feet long as shown in Figure 13. What is the standard form of the equation of the ellipse…
Com figura
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Algebra and Trigonometry
12.2 The Hyperbola
Exemplo 1
Locating a Hyperbola’s Vertices and Foci
Identify the vertices and foci of the hyperbola with equation (y^2)/(49) −(x^2)/(32) =1.
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Exemplo 2
Finding the Equation of a Hyperbola Centered at (0,0) Given its Foci and Vertices
What is the standard form equation of the hyperbola that has vertices ( ±6,0) and foci ( ±2√(10),0)?
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Exemplo 3
Finding the Equation of a Hyperbola Centered at (h, k) Given its Foci and Vertices
What is the standard form equation of the hyperbola that has vertices at (0,−2) and (6,−2) and foci at (−2,−2) and (8,−2)?
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Exemplo 4
Graphing a Hyperbola Centered at (0, 0) Given an Equation in Standard Form
Graph the hyperbola given by the equation (y^2)/(64) −(x^2)/(36) =1. Identify and label the vertices, co-vertices, foci, and asymptotes.
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Exemplo 5
Graphing a Hyperbola Centered at (h, k) Given an Equation in General Form
Graph the hyperbola given by the equation 9x^2 −4y^2 −36x−40y−388=0. Identify and label the center, vertices, co-vertices, foci, and asymptotes.
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Exemplo 6
Solving Applied Problems Involving Hyperbolas
The design layout of a cooling tower is shown in Figure 11. The tower stands 179.6 meters tall. The diameter of the top is 72 meters. At their closest, the sides of the tower are 60 meters apart. Figure 11 Project…
Com figura
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Algebra and Trigonometry
12.3 The Parabola
Exemplo 1
Graphing a Parabola with Vertex (0, 0) and the x-axis as the Axis of Symmetry
Graph y^2 =24x. Identify and label the focus, directrix, and endpoints of the latus rectum.
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Exemplo 2
Graphing a Parabola with Vertex (0, 0) and the y-axis as the Axis of Symmetry
Graph x^2 =−6y. Identify and label the focus, directrix, and endpoints of the latus rectum.
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Exemplo 3
Writing the Equation of a Parabola in Standard Form Given its Focus and Directrix
What is the equation for the parabola with focus ( −1/2,0) and directrix x=1/2?
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Exemplo 4
Graphing a Parabola with Vertex (h, k) and Axis of Symmetry Parallel to the x-axis
Graph ( y−1) ^2 =−16( x+3). Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the latus rectum.
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Exemplo 5
Graphing a Parabola from an Equation Given in General Form
Graph x^2 −8x−28y−208=0. Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the latus rectum.
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Exemplo 6
Solving Applied Problems Involving Parabolas
A cross-section of a design for a travel-sized solar fire starter is shown in Figure 13. The sun’s rays reflect off the parabolic mirror toward an object attached to the igniter. Because the igniter is located at the…
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Algebra and Trigonometry
12.4 Rotation of Axes
Exemplo 1
Identifying a Conic from Its General Form
Identify the graph of each of the following nondegenerate conic sections. ⓐ 4x^2 −9y^2 +36x+36y−125=0 ⓑ 9y^2 +16x+36y−10=0 ⓒ 3x^2 +3y^2 −2x−6y−4=0 ⓓ −25x^2 −4y^2 +100x+16y+20=0
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Exemplo 2
Finding a New Representation of an Equation after Rotating through a Given Angle
Find a new representation of the equation 2x^2 −xy+2y^2 −30=0 after rotating through an angle of θ=45°.
Exemplo 3
Rewriting an Equation with respect to the x′ and y′ axes without the x′y′ Term
Rewrite the equation 8x^2 −12xy+17y^2 =20 in the x^′ y^′ system without an x^′ y^′ term.
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Exemplo 4
Graphing an Equation That Has No x′y′ Terms
Graph the following equation relative to the x^′ y^′ system: x^2 +12xy−4y^2 =30
Exemplo 5
Identifying the Conic without Rotating Axes
Identify the conic for each of the following without rotating axes. ⓐ 5x^2 +2√3 xy+2y^2 −5=0 ⓑ 5x^2 +2√3 xy+12y^2 −5=0
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Algebra and Trigonometry
12.5 Conic Sections in Polar Coordinates
Exemplo 1
Identifying a Conic Given the Polar Form
For each of the following equations, identify the conic with focus at the origin, the directrix, and the eccentricity. r=6/(3+2sinθ) r=(12)/(4+5cosθ) r=7/(2−2sinθ)
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Exemplo 2
Graphing a Parabola in Polar Form
Graph r=5/(3+3cosθ).
Exemplo 3
Graphing a Hyperbola in Polar Form
Graph r=8/(2−3sinθ).
Exemplo 4
Graphing an Ellipse in Polar Form
Graph r=(10)/(5−4cosθ).
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Exemplo 5
Finding the Polar Form of a Vertical Conic Given a Focus at the Origin and the Eccentricity and Directrix
Find the polar form of the conic given a focus at the origin, e=3 and directrix y=−2.
Exemplo 6
Finding the Polar Form of a Horizontal Conic Given a Focus at the Origin and the Eccentricity and Directrix
Find the polar form of a conic given a focus at the origin, e=3/5, and directrix x=4.
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Exemplo 7
Converting a Conic in Polar Form to Rectangular Form
Convert the conic r=1/(5−5sinθ) to rectangular form.
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