Hyperbolas

11 Key excerpts on "Hyperbolas"

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  Intermediate Algebra

  Connecting Concepts through Applications

  • Mark Clark, Cynthia Anfinson(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Hyperbolas The last type of conic section we study is the hyperbola. Hyperbolas are formed when a plane slices through the upper and lower parts of joined cones. This occurs when the plane is parallel to the axis of the cones. The equation for a hyperbola is similar to that for an ellipse. However instead of a sum of two distances, there is a difference of two distances. A hyperbola is defined as the set of all points in a plane such that the difference of the distances from two fixed points (the foci) is constant. y x P g f Focus Focus The difference f 2 g is the same for every point P on the hyperbola. Hyperbola DEFINITION Geometric Definition of a Hyperbola The set of all points in a plane such that the difference of the distances from two fixed points is a constant is called a hyperbola. The standard equation for a hyperbola centered at the origin that opens right and left x 2 a 2 2 y 2 b 2 5 1 where a and b are positive real numbers. The foci are the points 1 c, 02 and ( 2c, 0), where c 2 5 a 2 1 b 2 , and the vertices are 1 a, 02 and 1 2a, 02 . The standard equation for a hyperbola centered at the origin that opens up and down y 2 b 2 2 x 2 a 2 5 1 where a and b are positive real numbers. The foci are the points 1 0, c 2 and (0, 2c), where c 2 5 a 2 1 b 2 , and the vertices are 1 0, b2 and 1 0, 2b2 . Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 798 The direction in which the hyperbola opens is determined by which variable term is positive. When the x 2 a 2 term is first, the hyperbola opens left and right and has x-intercepts at 1 a, 02 and ( 2a, 0).

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  Precalculus, Enhanced Edition

  • David Cohen, Theodore Lee, David Sklar, , David Cohen, Theodore Lee, David Sklar(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  —Theodore P. Snow in The Dynamic Universe: An Introduction to Astronomy, 4th ed. (St. Paul, Minn.: West Publishing Co., 1990) In the previous section we defined an ellipse as the set of points P such that the sum of the distances from P to two fixed points is constant. By considering the difference instead of the sum, we are led to the definition of the hyperbola. Definition The Hyperbola A hyperbola is the set of all points in the plane, the absolute value of the difference of whose distances from two fixed points is a positive constant. Each fixed point is called a focus. As with the ellipse, we label the foci F 1 and F 2 . Before obtaining an equation for the hyperbola, we can see the general features of the curve by using two sets of con-centric circles, with centers F 1 and F 2 , to locate points satisfying the definition of a hyperbola. In Figure 1 we’ve plotted a number of points P such that either F 1 P F 2 P 3 or F 2 P F 1 P 3. By joining these points, we obtain the graph of the hyperbola shown in Figure 1. Copyright 201 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Unlike the parabola or the ellipse, the hyperbola is composed of two distinct parts, or branches. As you can check, the left branch in Figure 1 corresponds to the equation F 2 P F 1 P 3, while the right branch corresponds to the equation F 1 P F 2 P 3. Figure 1 also reveals that the hyperbola possesses two types of symmetry. First, it is symmetric about the line passing through the two foci F 1 and F 2 ; this line is referred to as the focal axis of the hyperbola.

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  Precalculus: Mathematics for Calculus, International Metric Edition

  • James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Conic sections are the curves that are formed when a plane cuts a cone, as shown in the figure. For example, if a cone is cut horizontally, the cross section is a circle. So a circle is a conic section. Other ways of cutting a cone produce ellipses, parabolas, and Hyperbolas. Ellipse Parabola Hyperbola Circle Our goal in this chapter is to find equations whose graphs are conic sections. We will find such equations by analyzing the geometric properties of conic sections. These properties make conic sections useful for many real-world applications. For instance, a reflecting surface with parabolic cross sections concentrates light at a single point. This property of a parabola is used in the construction of solar power plants, like the one in California pictured above. In the Focus on Modeling at the end of the chapter we explore how these curves are used in architecture. 781 Conic Sections 11 11.1 Parabolas 11.2 Ellipses 11.3 Hyperbolas 11.4 Shifted Conics 11.5 Rotation of Axes 11.6 Polar Equations of Conics FOCUS ON MODELING Conics in Architecture Harald Sund/The Image Bank/Getty Images Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 782 CHAPTER 11 ■ Conic Sections 11.1 PARABOLAS ■ Geometric Definition of a Parabola ■ Equations and Graphs of Parabolas ■ Applications ■ Geometric Definition of a Parabola We saw in Section 3.1 that the graph of the equation y  ax 2  bx  c is a U-shaped curve called a parabola that opens either upward or downward, depend- ing on whether the number a is positive or negative.

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  Precalculus

  Functions and Graphs

  • Earl Swokowski, Jeffery Cole(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Conic sections were studied extensively by the ancient Greeks, who discovered properties that enable us to state their definitions in terms of points and lines, as we do in our discussion. From our work in Section 2.6, if a ± 0 , the graph of y 5 ax 2 1 bx 1 c is a parabola with a vertical axis. We shall next state a general definition of a parabola and derive equations for parabolas that have either a vertical axis or a horizontal axis. We shall assume that F is not on l , for this would result in a line. If P is a point in the plane and P 9 is the point on l determined by a line through P that is perpendicular to l (see Figure 2), then, by the preceding definition, P is on the parabola if and only if the distances d s P , F d and d s P , P 9 d are equal. The axis of the parabola is the line through F that is perpendicular to the directrix. The vertex of the parabola is the point V on the axis halfway from F to l . The vertex is the point on the parabola that is closest to the directrix. To obtain a simple equation for a parabola, place the y -axis along the axis of the parabola, with the origin at the vertex V , as shown in Figure 3. In this case, the focus F has coordinates s 0, p d for some real number p ± 0 , and the equation of the directrix is y 5 2 p . (The figure shows the case p . 0 .) By the FIGURE 2 P V Axis Directrix F P H11032 l Parabolas 10.1 Definition of a Parabola A parabola is the set of all points in a plane equidistant from a fixed point F (the focus ) and a fixed line l (the directrix ) that lie in the plane. 716 CHAPTER 10 Topics from Analytic Geometry Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.

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  Precalculus

  Building Concepts and Connections 2E

  • Revathi Narasimhan(Author)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  If the curve is closed, the conic is an ellipse; otherwise, it is a parabola. ■ If the plane is parallel to the axis of the cone and does not pass through the vertex, it intersects the cone in a pair of non-intersecting open curves, called a hyperbola. Each curve is called a branch of the hyperbola. In this section, we will define parabolas differently than we did in Chapter 2. Later on, we will make connections between the two definitions. Definition of a Parabola A parabola is the set of all points ( x , y ) in a plane such that the distance of ( x , y ) from a fixed line is equal to the distance of ( x , y ) from a fixed point that is not on the fixed line. The fixed line and the fixed point, which lie in the plane of the parabola, are called the directrix and the focus , respectively. The axis of symmetry of a parabola is the line that is perpendicular to the directrix and passes through the focus of the parabola. The vertex of a parabola is on the axis of symmetry, midway between the focus and the directrix. See Figure 2. The only parabolas discussed in this section are those whose axis of symmetry is either vertical or horizontal. Figure 2 Directrix Vertex Focus Axis of symmetry Objectives ■ Define a parabola. ■ Find the focus, directrix, and axis of symmetry of a parabola. ■ Determine the equation of a parabola and write it in standard form. ■ Translate a parabola in the xy -plane. ■ Sketch a parabola. ■ Understand the reflective property of a parabola. 738 Chapter 9 Conic Sections Parabolas with Vertex at the Origin Suppose a parabola has its vertex at the origin and the y -axis as its axis of symmetry. Suppose also that the equation of its directrix is y = − p , for some p > 0. The coordinates of the focus are (0, p ), since the vertex is midway between the focus and directrix. Let P be any point on the parabola with coordinates ( x , y ), and let dist( PF ) denote the distance from P to the focus F .

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  Precalculus

  Functions and Graphs, Enhanced Edition

  • Earl Swokowski, Jeffery Cole(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Conic sections were studied extensively by the ancient Greeks, who discovered properties that enable us to state their definitions in terms of points and lines, as we do in our discussion. From our work in Section 2.6, if , the graph of is a parabola with a vertical axis. We shall next state a general definition of a parabola and derive equations for parabolas that have either a vertical axis or a horizontal axis. We shall assume that F is not on l , for this would result in a line. If P is a point in the plane and is the point on l determined by a line through P that is perpendicular to l (see Figure 2), then, by the preceding definition, P is on the parabola if and only if the distances and are equal. The axis of the parabola is the line through F that is perpendicular to the directrix. The vertex of the parabola is the point V on the axis halfway from F to l . The vertex is the point on the parabola that is closest to the directrix. To obtain a simple equation for a parabola, place the y -axis along the axis of the parabola, with the origin at the vertex V , as shown in Figure 3. In this case, the focus F has coordinates for some real number , and the equation of the directrix is . (The figure shows the case .) By the p 0 y p p 0 0, p d P , P d P , F P y ax 2 bx c a 0 l Definition of a Parabola A parabola is the set of all points in a plane equidistant from a fixed point F (the focus ) and a fixed line l (the directrix ) that lie in the plane. FIGURE 2 P V Axis Directrix F P l Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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  Technical Mathematics with Calculus

  • Paul A. Calter, Michael A. Calter(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  It is used in the design of optical devices, highway curves, and much more. Definition of a Parabola We said that the parabola results when we cut a cone by a plane that is parallel to an element of the cone. Here is another definition. A parabola is the set of points in a plane, each of which is equidistant from a fixed point, the focus, 221 and a fixed line, the directrix. ■ Exploration: Try this. Use the preceding definition to construct a parabola whose distance from focus to directrix is 2.0 in. (a) Draw a line to represent the directrix as shown in Fig. 22–48, and indicate a focus F 2.0 in. from that line. (b) Then draw a line L parallel to the directrix at some arbitrary distance, say, 3.0 in. (c) With the same (3.0-in.) distance as radius and F as center, use a compass to draw arcs intersecting L at and Each of these points is now at the same distance (3.0 in.) from F and from the directrix and is hence a point on the parabola. (d) Repeat the construction with distances other than 3.0 in. to get more points on the parabola. ■ Figure 22–49 shows the typical shape of a parabola. The parabola has an axis of symmetry which intersects it at the vertex. The distance p from directrix to vertex is equal to the directed distance from the vertex to the focus. Standard Equation of a Parabola: Vertex at the Origin Let us place the parabola on coordinate axes with the vertex at the origin and with the axis of symmetry along the x axis, as shown in Fig. 22–50. Choose any point P on the parabola. Then, by the definition of a parabola, But in right triangle FBP, and But, since Squaring both sides yields Collecting terms, we get the standard equation of a parabola with vertex at the origin. 222 y 2  4px Standard Equation of a Parabola: Vertex at Origin, Axis Horizontal x 2  2px  p 2  y 2  p 2  2px  x 2 (x  p) 2  y 2  p 2  2px  x 2 4 (x  p) 2  y 2  p  x FP  AP, AP  p  x FP  4 (x  p) 2  y 2 FP  AP.

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  Geometry of Curves

  • J.W. Rutter(Author)
  • 2018(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  The axes of a hyperbola in canonical position coincide with the coordinate axes: for a hyperbola in general position this will not normally be the case. Hyperbolae in general position in space will have corresponding foci, and transverse and conjugate axes. A rectangular hyperbola is a hyperbola whose asymptotes (see below) are orthogonal: the asymptotes are orthogonal if, and only if, a = b. The hy-perbola differs from the ellipse in that it is an unbounded curve. Moreover it possess two asymptotes. Asymptotes are lines which, in a sense which will be made precise when we consider projective curves, are tangents to the curve 'at infinity'. Knowledge of asymptotes is of major importance in curve sketching and we will learn later how to determine them for more general curves. 1.5.5 Asymptotes The asymptotes of a hyperbola in canonical position are X V X V -= ± -, or equivalently — - — = 0. a b a 1 b z The axes of a hyperbola always bisect the asymptotes. A hyperbola is rectangular if, and only if, the asymptotes bisect the axes. We now sketch an (intuitive) proof that the asymptotes to the hyperbola in canonical position are as given above. A proof using the techniques of projective geometry will be given later (see §16.10.1) following our precise definition of asymptotes. x 2 w 2 Proof. The line y — mx 4-c meets the hyperbola — — — = 1 where a 2, b l _ m 2 2 _ 2 m c _£* r Ò 2 ) X b 2 X Ò 2 Both roots of the equation will be 'infinite' (and therefore 'equal') if the coefficients of x 2 and of x are zero: this is the case if, and only if, m = ± — a and c — 0. The result follows easily. D For m = ±-and other values of c the equation in the above proof a has one infinite and one finite solution; and for other values of m there are either two distinct real solutions, two equal solutions for x (this case may correspond to a tangent or to two distinct points of intersection with different values for y), or no real solution.

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  College Algebra

  • Sheldon Axler(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  Taking square roots then shows that y ≈ ± 4 3 x. Each branch of a hyperbola may appear to be shaped like a parabola, but these curves are not parabolas. As one notable difference, a parabola cannot have the behavior described in part (c) of the example above. Compare the following definition of the foci of a hyperbola with the definition earlier in this section of the foci of an ellipse. section 2.3 Quadratic Expressions and Conic Sections 87 Foci of a hyperbola The foci of a hyperbola are two points with the property that the dif- ference of the distances from the foci to a point on the hyperbola is a constant independent of the point on the hyperbola. As we will see in the next example, the foci for the hyperbola y 2 16 - x 2 9 = 1 Problems 81–84 show why the graph of y = 1 x is also called a hyperbola. are the points (0, -5) and (0, 5). This example shows how to verify that a pair of points are foci for a hyperbola. example 12 (a) Find a formula in terms of y for the distance from a typical point (x, y) with y > 0 on the hyperbola y 2 16 - x 2 9 = 1 to the point (0, -5). (b) Find a formula in terms of y for the distance from a typical point (x, y) with y > 0 on the hyperbola y 2 16 - x 2 9 = 1 to the point (0, 5). (c) Show that (0, -5) and (0, 5) are foci of the hyperbola y 2 16 - x 2 9 = 1. solution (a) The distance from (x, y) to the point (0, -5) is q x 2 + (y + 5) 2 , which equals q x 2 + y 2 + 10y + 25. We want an answer solely in terms of y , assuming that (x, y) lies on the hyperbola y 2 16 - x 2 9 = 1 and y > 0. Solving the hyperbola equation for x 2 , we have x 2 = 9 16 y 2 - 9. Substituting this expression for x 2 into the expression above shows that the distance from (x, y) to (0, -5) equals q 25 16 y 2 + 10y + 16, which equals r ( 5 4 y + 4) 2 , which equals 5 4 y + 4. x ,y  6 6 x 5 5 4 4 y For every point (x, y) on the hyperbola y 2 16 - x 2 9 = 1, the difference of the lengths of the two red line segments equals 8.

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  Algebra II Workbook For Dummies

  • Mary Jane Sterling(Author)
  • 2018(Publication Date)
  • For Dummies

    (Publisher)

  Chapter 10

  Any Way You Slice It: Conic Sections

  IN THIS CHAPTER

  Writing equations of parabolas, circles, ellipses, and Hyperbolas in standard form

  Identifying centers, foci, and axes of conic sections

  Sketching parabolas, circles, ellipses, and Hyperbolas

  Chapter 6 contains information on quadratic functions, which you graph as U-shaped curves called parabolas. Here, you see that not all parabolas represent functions. When a parabola opens to the left or right, you get more than one y -value for many x -values, and that just doesn’t work with functions. This chapter introduces parabolas and some other conic sections that don’t exactly fit the function mold. All parabolas are conic sections, but not all parabolas are functions — it depends on how they lie.

  A conic section, or conic, is represented by a slice of a cone (or really, two cones stacked on top of each other, like an hourglass). How you slice the cone — at what angle and how big a chunk you take — determines what kind of conic section you produce. The four types of conic sections are the parabola, circle, ellipse, and hyperbola. Here’s how they compare:

  • Parabola: You get this U -shaped curve when you take a slice through a cone down to the base.
  • Circle: A circle appears when you cut a cone straight across, parallel to the base.
  • Ellipse: If you slice through a cone at a slant, you get an ellipse, or oval.
  • Hyperbola: Slicing straight down through both cones gives you a hyperbola. It looks like two U -shaped curves facing away from each other.

  In this chapter, you use the standard forms of equations of conics to recognize which type of conic section an equation represents. You also find the defining characteristics of the conic sections and use these traits to help you graph the conics.

  Putting Equations of Parabolas in Standard Form

  Rules representing parabolas come in two standard forms to separate the functions opening upward or downward from relations

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  Geometry

  • David A. Brannan, Matthew F. Esplen, Jeremy J. Gray(Authors)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  parabola and a hyperbola, respectively. This completes the proof of our claim in Subsection 1.1.1 that the curves of intersection of certain planes with a double cone are an ellipse, a parabola or a hyperbola. We shall investigate the converse in Theorem 4 of Subsection 4.1.4 . 1.2 Properties of Conics 1.2.1 Tangents In the previous section you met the parametric equations of the parabola, Subsection 1.1.3 ellipse and hyperbola in standard form. We now tackle a rather natural question: given parametric equations x = curve tangent ( x ( t ), y ( t )) x ( t ) , y = y ( t ) describing a curve, what is the slope of the tangent to the curve at the point with parameter t ? This information will enable us to determine the equation of the tangent to the curve at that point. ( x ( t ), y ( t )) ( x ( t + h ), y ( t + h )) y ( t + h ) – y ( t ) y ( t + h ) – x ( t ) Theorem 1 The slope of the tangent to a curve in R 2 with parametric equations x = x ( t ) , y = y ( t ) at the point with parameter t is y ( t ) x ( t ) , provided that x ( t ) = 0. Proof The points on the curve with parameters t and t + h have coordinates ( x ( t ) , y ( t )) and ( x ( t + h ) , y ( t + h )) , respectively. Then, if h = 0, the slope of the chord joining these two points is y ( t + h ) − y ( t ) x ( t + h ) − x ( t ) , which we can write in the form ( y ( t + h ) − y ( t ))/ h ( x ( t + h ) − x ( t ))/ h . 24 1: Conics We then take the limit of this ratio as h → 0. The slope of the chord tends to the slope of the tangent, namely y ( t )/ x ( t ) . Example 1 (a) Determine the equation of the tangent at the point with parameter t to the –a –b b y x a ellipse with parametric equations x = a cos t , y = b sin t , where t ∈ ( − π , π ] , t = 0, π . (b) Hence determine the equation of the tangent to the ellipse with parametric equations x = 3 cos t , y = sin t at the point with parameter t = π/ 4. Deduce the coordinates of the point of intersection of this tangent with the x -axis.

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