Parametric Hyperbolas

6 Key excerpts on "Parametric Hyperbolas"

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  Single Variable Calculus, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  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. 744 CHAPTER 10 Parametric Equations and Polar Coordinates (see Exercise 67). This principle is used in lithotripsy, a treatment for kidney stones. A reflector with elliptical cross-section is placed in such a way that the kidney stone is at one focus. High-intensity sound waves generated at the other focus are reflected to the stone and destroy it without damaging surrounding tissue. The patient is spared the trauma of surgery and recovers within a few days. ■ Hyperbolas A hyperbola is the set of all points in a plane the difference of whose distances from two fixed points F 1 and F 2 (the foci) is a constant. This definition is illustrated in Fig- ure 11. Hyperbolas occur frequently as graphs of equations in chemistry, physics, biology, and economics (Boyle’s Law, Ohm’s Law, supply and demand curves). A particularly significant application of hyperbolas was found in the long-range navigation systems developed in World Wars I and II (see Exercise 53). Notice that the definition of a hyperbola is similar to that of an ellipse; the only change is that the sum of distances has become a difference of distances. In fact, the derivation of the equation of a hyperbola is also similar to the one given earlier for an ellipse. It is left as Exercise 54 to show that when the foci are on the x-axis at s6c, 0d and the difference of distances is | PF 1 | 2 | PF 2 | - 62a, then the equation of the hyperbola is 6 x 2 a 2 2 y 2 b 2 - 1 where c 2 - a 2 1 b 2 . Notice that the x-intercepts are again 6a and the points sa, 0d and s2a, 0d are the vertices of the hyperbola.

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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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  Anton's Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  650 Chapter 10 / Parametric and Polar Curves; Conic Sections Hyperbola with center (h, k) and focal axis parallel to y-axis (y − k) 2 a 2 − (x − h) 2 b 2 = 1 (19) Example 7 Find an equation for the parabola that has its vertex at (1, 2) and its focus at (4, 2). Solution. Since the focus and vertex are on a horizontal line, and since the focus is to the right of the vertex, the parabola opens to the right and its equation has the form (y − k) 2 = 4p(x − h) Since the vertex and focus are 3 units apart, we have p = 3, and since the vertex is at (h, k) = (1, 2), we obtain (y − 2) 2 = 12(x − 1) Sometimes the equations of translated conics occur in expanded form, in which case we are faced with the problem of identifying the graph of a quadratic equation in x and y: Ax 2 + Cy 2 + Dx + Ey + F = 0 (20) The basic procedure for determining the nature of such a graph is to complete the squares of the quadratic terms and then try to match up the resulting equation with one of the forms of a translated conic. Example 8 Describe the graph of the equation y 2 − 8x − 6y − 23 = 0 Solution. The equation involves quadratic terms in y but none in x, so we first take all of the y-terms to one side: y 2 − 6y = 8x + 23 Next, we complete the square on the y-terms by adding 9 to both sides: ( y − 3) 2 = 8x + 32 Finally, we factor out the coefficient of the x-term to obtain ( y − 3) 2 = 8(x + 4) This equation is of form (12) with h = −4, k = 3, and p = 2, so the graph is a parabola with vertex (−4, 3) opening to the right. Since p = 2, the focus is 2 units to the right of the vertex, which places it at the point (−2, 3); and the directrix is 2 units to the left of the vertex, which means that its equation is x = −6. The parabola is shown in Figure 10.4.27. x y (−4, 3) (−2, 3) Directrix x = −6 y 2 − 8x − 6y − 23 = 0 Figure 10.4.27 Example 9 Describe the graph of the equation 16x 2 + 9y 2 − 64x − 54y + 1 = 0 Solution.

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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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  Mathematical Techniques in GIS

  • Peter Dale(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  Its parametric form uses the fact that (1 + tan 2 θ ) = (1 + sin 2 θ /cos 2 θ ) = (cos 2 θ + sin 2 θ )/cos 2 θ = sec 2 θ . Thus, sec 2 θ – tan 2 θ = 1 The hyperbola can therefore be expressed in the form x = a sec θ , y = b tan θ or using the parameter t x = a sec t ; y = b tan t Parametric forms (see Box 9.1) can be used to determine the slope at any point on a curve. If, for example, we consider the circle with center ( x c ,y c ) and radius r , then its parametric form is: x = x c + r cos t y = y c + r sin t Differentiating with respect to t d x /d t = – r sin t ; d y /d t = r cos t (d y /d t )/(d x /d t ) = –(cos t /sin t ) = –cot t BOX 9.1 PARAMETRIC EQUATIONS FOR LINES AND CONIC SECTIONS (SECOND-DEGREE CURVES) p , q , and θ are constant, that is, fixed quantities; t is the independent variable. For a line: x = p + t cos θ ; y = q + t sin θ For a circle: x = p + r cos t ; y = q + r sin t For an ellipse: x = p + a cos t ; y = q + b sin t For a parabola: x = p + at 2 ; y = q + 2 at For a hyperbola: x = p + a sec t ; y = q + b tan t 186 Mathematical Techniques in GIS or d y /d x = –cot t This is the rate of change of the curve, or put another way, it is the measure of the slope of the curve at the point defined by t . 9.2 THE ELLIPSE Figure 9.4 shows the auxiliary circle of radius a and the ellipse that has semiminor axis of length b . A point P on the ellipse has coordinates ( a cos θ , b sin θ ) referred to the center. PT is the tangent to the ellipse at point P with T being on the major axis. PQ is the normal at P , with Q being on the minor axis. The line QP makes an angle ø with the major axis and is known as the geodetic or spheroidal latitude of P . It is essentially the direction of the vertical at P . In practice, a plumb bob on the Earth’s surface may not point exactly in this direction because of the gravitational attraction of nearby mountains, giving rise to what is known in geodesy as the deviation of the vertical .

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  Calculus Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  620 Chapter 9 / Parametric and Polar Curves; Conic Sections 20 ft 12 ft 12 ft 40 ft ▶ Figure Ex-31 32. a. Find an equation for the parabolic arch with base b and height h, shown in the accompanying figure. b. Find the area under the arch. x y (b, 0) ( b, h) 1 2 ▶ Figure Ex-32 33. Show that the vertex is the closest point on a parabola to the focus. [Suggestion: Introduce a convenient coordinate system and use Definition 9.4.1.] 34. As illustrated in Figure Ex-34, suppose that a comet moves in a parabolic orbit with the Sun at its focus and that the line from the Sun to the comet makes an angle of 30 ◦ with the axis of the parabola when the comet is 10 √ 3 million miles from the center of the Sun. Use the result in Exercise 33 to determine how close the comet will come to the center of the Sun. 35. For the parabolic reflector in the accompanying figure, how far from the vertex should the light source be placed to produce a beam of parallel rays? 60° ▴ Figure Ex-34 1 ft 1 ft ▴ Figure Ex-35 36. a. Show that the right and left branches of the hyperbola x 2 a 2 − y 2 b 2 = 1 can be represented parametrically as x = a cosh t, y = b sinh t (−∞ < t < + ∞) x = −a cosh t, y = b sinh t (−∞ < t < + ∞) b. Use a graphing utility to generate both branches of the hyperbola x 2 − y 2 = 1 on the same screen. 37. a. Show that the right and left branches of the hyperbola x 2 a 2 − y 2 b 2 = 1 can be represented parametrically as x = a sec t, y = b tan t (−∕2 < t < ∕2) x = −a sec t, y = b tan t (−∕2 < t < ∕2) b. Use a graphing utility to generate both branches of the hyperbola x 2 − y 2 = 1 on the same screen. 38. An ant is walking on the xy-plane is such a way that its distance to the point (1, 1) is always the same as its distance to the x-axis. Find an equation for the path of the ant. 39. Find an equation of the parabola traced by a point that moves so that its distance from (3, 6) is the same as its distance to the x-axis.

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