| 单词 | hyperbola |
| 例句 | Take a cone and cut it up; you get circles, ellipses, parabolas, and hyperbolas, depending on how you slice it. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z But Newton was the first to propose an exact mathematical form and to use that form to show that the motion of heavenly bodies should be conic sections—circles, ellipses, parabolas, and hyperbolas. College Physics for AP Courses 2015-08-12T00:00:00Z Its orbit is a hyperbola, and when discovered it was already rapidly leaving the inner solar system. Astronomy 2016-10-13T00:00:00Z A hyperbola is all points in a plane where the difference of their distances from two fixed points is constant. College Algebra with Corequisite Support 2020-09-23T00:00:00Z If eccentricity is greater than 1, it is a hyperbola. College Algebra 2021-12-21T00:00:00Z Each of the fixed points is called a focus of the hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z The hyperbolas represent ideal-gas behavior at various fixed temperatures, and are called isotherms. College Physics for AP Courses 2015-08-12T00:00:00Z The circle, ellipse, parabola, and hyperbola are all formed by the intersection of a plane with a cone. Astronomy 2016-10-13T00:00:00Z Figure 1.51 A typical hyperbola in which the difference of the distances from any point on the ellipse to the foci is constant. Calculus, Volume 3 2016-03-30T00:00:00Z The foci must lie on the transverse axis and be in the interior of the hyperbola. College Algebra 2021-12-21T00:00:00Z The midpoint of the segment joining the foci is called the center of the hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z At lower temperatures, the curves begin to look less like hyperbolas—the gas is not behaving ideally and may even contain liquid. College Physics for AP Courses 2015-08-12T00:00:00Z The graph of this function is called a rectangular hyperbola as shown. Calculus, Vol 2 2016-03-30T00:00:00Z The eccentricity of an ellipse is less than 1, the eccentricity of a parabola is equal to 1, and the eccentricity of a hyperbola is greater than 1. Calculus, Volume 3 2016-03-30T00:00:00Z We may wish to write the rectangular equation in the hyperbola’s standard form. Algebra and Trigonometry 2015-02-13T00:00:00Z In this chapter, you will learn about conics, including circles, parabolas, ellipses, and hyperbolas. Intermediate Algebra 2020-05-06T00:00:00Z "Not every hyperbola is a burial or a grave — there's tree roots, there's masses, and other things in the substrate," Small says. Radar is tracking down missing indigenous grave sites 2022-04-13T04:00:00Z In the case of a hyperbola, there are two foci and two directrices. Calculus, Vol 2 2016-03-30T00:00:00Z For the following exercises, determine the equation of the hyperbola using the information given. Calculus, Volume 3 2016-03-30T00:00:00Z If it is a hyperbola, label the vertices and foci. Algebra and Trigonometry 2015-02-13T00:00:00Z We define a hyperbola as all points in a plane where the difference of their distances from two fixed points is constant. Intermediate Algebra 2020-05-06T00:00:00Z But most of the world of commodities is defined by hyperbola- or parabola-shaped curves. What's the Deal With Oil Prices? 2016-02-02T05:00:00Z The derivation of the equation of a hyperbola in standard form is virtually identical to that of an ellipse. Calculus, Vol 2 2016-03-30T00:00:00Z It is a hyperbola if e > 1, a parabola if e = 1, or an ellipse if e < 1. Calculus, Volume 3 2016-03-30T00:00:00Z For the following exercises, graph the hyperbola, labeling vertices and foci. Algebra and Trigonometry 2015-02-13T00:00:00Z Each of the fixed points is called a focus of the hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z One of the portions of a curve that extends outwards to an indefinitely great distance; as, the branches of an hyperbola. Webster's Unabridged Dictionary (2nd 100 Pages) 2012-03-24T02:00:19.387Z The equation of a hyperbola is in general form if it is in the form Ax2 + By2 + Cx + Dy + E = 0, where A and B have opposite signs. Calculus, Vol 2 2016-03-30T00:00:00Z Then the definition of the hyperbola gives d⎛P, F ⎞ − d⎛P, F ⎞ = constant. Calculus, Volume 3 2016-03-30T00:00:00Z If it is an ellipse or a hyperbola, label the vertices and foci. Algebra and Trigonometry 2015-02-13T00:00:00Z The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z The two axes of the ellipse are the major axis and the minor axis, and the two axes of the hyperbola are the transverse axis and the conjugate axis. Webster's Unabridged Dictionary (2nd 100 Pages) 2012-03-24T02:00:19.387Z This property of the hyperbola has important applications. Calculus, Vol 2 2016-03-30T00:00:00Z We can view these surfaces as three-dimensional extensions of the conic sections we discussed earlier: the ellipse, the parabola, and the hyperbola. Calculus, Volume 3 2016-03-30T00:00:00Z The distinction is that the hyperbola is defined in terms of the difference of two distances, whereas the ellipse is defined in terms of the sum of two distances. Algebra and Trigonometry 2015-02-13T00:00:00Z The midpoint of the segment joining the foci is called the center of the hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z The epithets hyperbolic and parabolic are of course derived from the conic hyperbola and parabola respectively. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 8 "Cube" to "Daguerre, Louis" 2012-01-31T03:00:17.257Z The asymptotes of this hyperbola are the x and y coordinate axes. Calculus, Vol 2 2016-03-30T00:00:00Z Therefore, any points on the hyperbola are not only critical points, they are also on the boundary of the domain. Calculus, Volume 3 2016-03-30T00:00:00Z As with the ellipse, every hyperbola has two axes of symmetry. Algebra and Trigonometry 2015-02-13T00:00:00Z Each piece of the graph is called a branch of the hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z The nature of the two kinds of branches is best understood by considering them as projections, in the same way as we in effect consider the hyperbola and the parabola as projections of the ellipse. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 8 "Cube" to "Daguerre, Louis" 2012-01-31T03:00:17.257Z A graph of a typical hyperbola appears as follows. Calculus, Vol 2 2016-03-30T00:00:00Z Their under surfaces were of many different designs, and were of as many shapes as it is conceivable could be made of combinations of the circle and hyperbola. Etidorhpa or the End of Earth. The Strange History of a Mysterious Being and The Account of a Remarkable Journey 2011-10-18T02:00:20.997Z A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone. Algebra and Trigonometry 2015-02-13T00:00:00Z So, the equation of a hyperbola centered at the origin in standard form is: Intermediate Algebra 2020-05-06T00:00:00Z It is natural, therefore, that circle, ellipse, parabola, and hyperbola should all be looked upon as lines. The Teaching of Geometry 2011-10-12T02:00:52.133Z Figure 7.51 A typical hyperbola in which the difference of the distances from any point on the ellipse to the foci is constant. Calculus, Vol 2 2016-03-30T00:00:00Z The axis of a curved line is formed by a right line dividing the curve into two symmetrical parts, as in the parabola, ellipse, and hyperbola:—pl. Chambers's Twentieth Century Dictionary (part 1 of 4: A-D) 2011-10-11T02:01:08.990Z The graph of this equation is a hyperbola. Algebra and Trigonometry 2015-02-13T00:00:00Z The a, b values in the equation also help us find the asymptotes of the hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z If the plane tips the slightest amount more, the section becomes an hyperbola. The Teaching of Geometry 2011-10-12T02:00:52.133Z The graph of this hyperbola appears in the following figure. Calculus, Vol 2 2016-03-30T00:00:00Z From the line-pair we pass through an infinity of hyperbolas to the parabola, and thence through an infinity of ellipses to the circle. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The standard form of a hyperbola can be used to locate its vertices and foci. Algebra and Trigonometry 2015-02-13T00:00:00Z We will use these properties to graph hyperbolas. Intermediate Algebra 2020-05-06T00:00:00Z Two of the sides of the triangle in this proposition constitute a special form of the hyperbola. The Teaching of Geometry 2011-10-12T02:00:52.133Z That the protest was made, that it was sincere, rational and free from the hyperbola of passion, is the significant fact. Socialism and Democracy in Europe 2011-03-15T02:00:12.887Z The circle has one focus at the centre, an ellipse or hyperbola two foci equidistant from the centre. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form. Algebra and Trigonometry 2015-02-13T00:00:00Z Since the y2 - term is positive, the hyperbola opens up and down. Intermediate Algebra 2020-05-06T00:00:00Z But after you have demonstrated to him the properties of the hyperbola and its asymptote, the apparent absurdity vanishes. The Religion of Geology and Its Connected Sciences 2011-02-28T03:00:28.890Z A body moving in a parabola or hyperbola would recede indefinitely from its centre of motion and never return to it. Encyclopaedia Britannica, 11th Edition, Volume 2, Slice 7 "Arundel, Thomas" to "Athens" Lines being some positive and some negative, there must be negative rectangles and negative squares, such as those of the exterior diameters of a hyperbola. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola. Algebra and Trigonometry 2015-02-13T00:00:00Z Since the x2 - term is positive, the hyperbola opens left and right. Intermediate Algebra 2020-05-06T00:00:00Z The vertex was, however, in reality rounded off, and the figure rather resembled a hyperbola, which tended to coincidence with its asymptotes. The Glaciers of the Alps Being a narrative of excursions and ascents, etc. If the second medium be more highly refractive than the first, the secondary caustic is a hyperbola having the same focus and centre as before, and the caustic is the evolute of this curve. Encyclopaedia Britannica, 11th Edition, Volume 5, Slice 5 "Cat" to "Celt" Let CC be the other foci of the ellipse and the hyperbola. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Real-world situations can be modeled using the standard equations of hyperbolas. Algebra and Trigonometry 2015-02-13T00:00:00Z Draw a sketch of the hyperbola labeling the center, vertices, and asymptotes. Intermediate Algebra 2020-05-06T00:00:00Z The apex of each being thrust forward, they take the form of sharp hyperbolas, and preserve this character throughout the entire length of the Mer de Glace. The Glaciers of the Alps Being a narrative of excursions and ascents, etc. When the conjugate axis of the hyperbola is made smaller and smaller, the nodoid approximates more and more to the series of spheres touching each other along the axis. Encyclopaedia Britannica, 11th Edition, Volume 5, Slice 3 "Capefigue" to "Carneades" Then any point of the ellipse is equidistant from the focus B and one circle, and any point of the hyperbola from the focus B and the other circle. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z They include an ellipse, a circle, a hyperbola, and a parabola. Algebra and Trigonometry 2015-02-13T00:00:00Z Explain in your own words how to create and use the rectangle that helps graph a hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z In such cases the orbit might be changed to a hyperbola, and then the comet would never return. Encyclopaedia Britannica, 11th Edition, Volume 6, Slice 7 "Columbus" to "Condottiere" It is said that he was the first to introduce the words ellipse and hyperbola. History of the Intellectual Development of Europe, Volume I (of 2) Revised Edition This gives— An ellipse as well as an hyperbola has one pair of axes. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z For the following exercises, find the equation of the hyperbola. Algebra and Trigonometry 2015-02-13T00:00:00Z With systems of nonlinear equations, the graphs may be circles, parabolas or hyperbolas and there may be several points of intersection, and so several solutions. Intermediate Algebra 2020-05-06T00:00:00Z Consequently, anything coming to the Sun from a great distance must have one exact amount of proper motion to produce a parabola: all other amounts would give hyperbolas or ellipses. Essays: Scientific, Political, & Speculative, Vol. I No comet has yet been observed to travel in a decided hyperbola. A Popular History of Astronomy During the Nineteenth Century Fourth Edition For the asymptotes as axes of co-ordinates the equation of the hyperbola is xy = const. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z Sketch the possible options for intersection of a circle and hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z Ellipses and hyperbolas have countless varieties of forms, but there is only one form of parabola; or, to speak literally, all parabolas are similar, while there are infinitely numerous dissimilar ellipses and dissimilar hyperbolas. Essays: Scientific, Political, & Speculative, Vol. I It is manifest that the plane will thus intersect the higher cone, and produce the figure which is known to mathematicians as the hyperbola. The Romance of Mathematics Being the Original Researches of a Lady Professor of Girtham College in Polemical Science, with some Account of the Social Properties of a Conic; Equations to Brain Waves; Social Forces; and the Laws of Political Motion. The first may also be enunciated thus— The lines which join any point on an ellipse or an hyperbola to the ends of a diameter are parallel to two conjugate diameters. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z For the following exercises, graph the hyperbola, noting its center, vertices, and foci. Algebra and Trigonometry 2015-02-13T00:00:00Z The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z By continuing this process, the ellipse passes insensibly into a parabola; and, ultimately, by still further diminishing the angle, into an hyperbola. Essays: Scientific, Political, & Speculative, Vol. I Beyond them the lower slopes of the Devil's Peak are yellow and red sand, but the grey-green waters of the bay, which is shaped like a great hyperbola, are edged with white sand. A Tramp's Notebook This reasoning shows at the same time how to construct the axis of an ellipse or of an hyperbola. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z If it is an ellipse or a hyperbola, label vertices and foci. Algebra and Trigonometry 2015-02-13T00:00:00Z Each piece of the graph is called a branch of the hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z E.g., in the case of the hyperbola xy = 1, the area between x = a and x = 1 is log a. A Budget of Paradoxes, Volume I Menaechmus, a pupil of Eudoxus, was the discoverer of the conic sections, two of which, the parabola and the hyperbola, he used for solving the problem of the two mean proportionals. The Legacy of Greece Essays By: Gilbert Murray, W. R. Inge, J. Burnet, Sir T. L. Heath, D'arcy W. Thompson, Charles Singer, R. W. Livingston, A. Toynbee, A. E. Zimmern, Percy Gardner, Sir Reginald Blomfield If the chord is changed into a tangent, this gives— The segment between the asymptotes on any tangent to an hyperbola is bisected by the point of contact. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z A hyperbola is the set of all points in a plane such that the difference of the distances between and the foci is a positive constant. Algebra and Trigonometry 2015-02-13T00:00:00Z There are four conics—the circle, parabola, ellipse, and hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z The parabolic curve, as it is called, is a line partaking of the closeness of the ellipse on the one hand, and the openness of the hyperbola on the other. Chambers's Edinburgh Journal, No. 450 Volume 18, New Series, August 14, 1852 If the cone is cut off vertically on the dotted line, A, the curve is a hyperbola. Carpentry for Boys In a Simple Language, Including Chapters on Drawing, Laying Out Work, Designing and Architecture With 250 Original Illustrations The centre of an hyperbola is without the curve, because the line at infinity cuts the curve. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Notice that the definition of a hyperbola is very similar to that of an ellipse. Algebra and Trigonometry 2015-02-13T00:00:00Z We will see that the equation of a hyperbola looks the same as the equation of an ellipse, except it is a difference rather than a sum. Intermediate Algebra 2020-05-06T00:00:00Z In the hyperbola we have the mathematical demonstration of the error of an axiom. Fables of Infidelity and Facts of Faith Being an Examination of the Evidences of Infidelity But as regards the fact, it is probable that Mr. Walker’s views are correct, so far as the change from an ellipse to an hyperbola is considered. Outlines of a Mechanical Theory of Storms Containing the True Law of Lunar Influence In case of the ellipse and hyperbola, the pole to the line at infinity is a finite point called the centre of the curve. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The transverse axis is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. Algebra and Trigonometry 2015-02-13T00:00:00Z While the equations of an ellipse and a hyperbola are very similar, their graphs are very different. Intermediate Algebra 2020-05-06T00:00:00Z It is relatively like what is said of a straight line drawn about a hyperbola, always approaching but never touching it, and like what is said about squaring a circle. Angelic Wisdom about Divine Providence If F is within the circle, the conic will be an ellipse; if F is without the circle, the conic will be a hyperbola. An Elementary Course in Synthetic Projective Geometry Hence also— From the centre of an hyperbola two tangents can be drawn to the curve which have their point of contact at infinity. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The central rectangle of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z In your own words, define a hyperbola and write the equation of a hyperbola centered at the origin in standard form. Intermediate Algebra 2020-05-06T00:00:00Z It may prove just as unreliable as the hyperbola. All Around the Moon Apply the theorem of problem 9 to the case of a hyperbola where the two tangents are the asymptotes. An Elementary Course in Synthetic Projective Geometry Hence— Any two conjugate diameters of an hyperbola are harmonic conjugates with regard to the asymptotes. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The hyperbola is the set of all points such that the difference of the distances from to the foci is constant. Algebra and Trigonometry 2015-02-13T00:00:00Z There are also four options when we consider a circle and a hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z "I only want to make it clear as day that it is hyperbola!" All Around the Moon By taking the point at the vertex of the conic, we note that this ratio is less than unity for the ellipse, greater than unity for the hyperbola, and equal to unity for the parabola. An Elementary Course in Synthetic Projective Geometry Two asymptotes and any two tangents to an hyperbola may be considered as a quadrilateral circumscribed about the 701 hyperbola. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z If is a point on the hyperbola, we can define the following variables: Algebra and Trigonometry 2015-02-13T00:00:00Z The lines containing the diagonals of this rectangle are the asymptotes of the hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z Then let the Eye of Science and her parabolas and hyperbolas, and conjugates, and asymptotes, and the rest of the confounded nonsensical farrago, all go to pot! All Around the Moon The tangent AB may be fixed, and the tangent CD chosen arbitrarily; therefore The triangle formed by any tangent to the hyperbola and the two asymptotes is of constant area. An Elementary Course in Synthetic Projective Geometry If we take instead of P the centre C of an hyperbola, then the conjugate lines become conjugate diameters, and the tangents asymptotes. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z We know that the difference of these distances is for the vertex It follows that for any point on the hyperbola. Algebra and Trigonometry 2015-02-13T00:00:00Z The rectangle and asymptotes are not part of the hyperbola, but they help us graph the hyperbola. Intermediate Algebra 2020-05-06T00:00:00Z "Exactly so," chimed in M'Nicholl; "it must describe and keep on describing either a parabola or a hyperbola." All Around the Moon This identifies the curve with the hyperbola as defined and discussed in works on analytic geometry. An Elementary Course in Synthetic Projective Geometry The first part allows a simple solution of the problem to find any number of points on an hyperbola, of which the asymptotes and one point are given. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Label the foci and asymptotes, and draw a smooth curve to form the hyperbola, as shown in Figure 8. Algebra and Trigonometry 2015-02-13T00:00:00Z Sometimes the equation for a hyperbola needs to be first placed in standard form before we graph it. Intermediate Algebra 2020-05-06T00:00:00Z On the inner part of the cover are marked in the first place the horary lines, traversed by curves that are symmetrical with respect to the vertical and having the aspect of arcs of hyperbolas. Scientific American Supplement, No. 810, July 11, 1891 Segments cut off on a chord by hyperbola and its asymptotes116. An Elementary Course in Synthetic Projective Geometry From this the following theorem is a simple deduction: All triangles formed by a tangent and the asymptotes of an hyperbola are equal in area. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z TRY IT #4 Graph the hyperbola given by the equation Identify and label the vertices, co- vertices, foci, and asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z Further on, the wall ends at the foot of a tower called Quiquengrogne; like its sister, La Générale, it is high, broad, and imposing, and is swelled in the middle like a hyperbola. Over Strand and Field The following analysis shows that with the aid of an hyperbola any arc, and therefore any angle, may be trisected. Scientific American Supplement, No. 787, January 31, 1891 But in order to render it adjustable, so that hyperbolas of varying eccentricities and on different scales may be drawn with it, several parts not here shown must be added. Scientific American Supplement, No. 530, February 27, 1886 The axis which contains the foci is called the principal axis; in case of an hyperbola it is the axis which cuts the curve, because the foci lie within the conic. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z To do this, we can use the dimensions of the tower to find some point that lies on the hyperbola. Algebra and Trigonometry 2015-02-13T00:00:00Z One was for the hyperbola, the other for the parabola. The Moon-Voyage The body under the circumstances might describe an hyperbola as welt as an ellipse, as Professor Mitchell himself subsequently remarks. The Atlantic Monthly, Volume 06, No. 33, July, 1860 The form of this melted part could be made to differ considerably, but approximated to that of an equilateral hyperbola. Scientific American Supplement, No. 415, December 15, 1883 The branches of the hyperbola approach these lines indefinitely as a point on the curves moves to infinity. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Find the equation of the hyperbola that models the sides of the cooling tower. Algebra and Trigonometry 2015-02-13T00:00:00Z "Yes," answered Barbicane, "with some speed it will describe a parabola, and with greater speed an hyperbola." The Moon-Voyage Little by little, the mountains will be worn down to a surface of less and less relief, approaching a plain as a hyperbola approaches its asymptote—a surface which W. M. Davis has called peneplain. Popular Science Monthly Oct, Nov, Dec, 1915 — Volume 86 "Yes," said Nicholl, "it will follow either a parabola or a hyperbola." From the Earth to the Moon; and, Round the Moon The surface is thus generated by a variable hyperbola moving parallel to itself along the parabolas as directrices. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Define a hyperbola in terms of its foci. Algebra and Trigonometry 2015-02-13T00:00:00Z I say, Messrs. Cosine, do leave off throwing your hyperbolas and parabolas at one's head. The Moon-Voyage If the cutting plane be produced so as to cut the opposite cone, another curve will be formed, which is also an hyperbola. The Gutenberg Webster's Unabridged Dictionary Section F, G and H Now, gentlemen cosines, will you cease to throw parabolas and hyperbolas at each other's heads? From the Earth to the Moon; and, Round the Moon The hyperbola would have an equation of the form xy = c if referred to its asymptotes as axes, the coordinates being then oblique, unless a = b, in which case the hyperbola is called rectangular. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z What can we conclude about a hyperbola if its asymptotes intersect at the origin? Algebra and Trigonometry 2015-02-13T00:00:00Z "Yes, it will either describe a parabola or an hyperbola." The Moon-Voyage Both curves are regarded as branches of the same hyperbola. The Gutenberg Webster's Unabridged Dictionary Section F, G and H "Then, no more hyperbolas, no more parabolas, no more open curves to fear?" From the Earth to the Moon; and, Round the Moon The conic sections are the parabola, ellipse, and hyperbola. The Gutenberg Webster's Unabridged Dictionary Section C Where must the center of hyperbola be relative to its foci? Algebra and Trigonometry 2015-02-13T00:00:00Z Barbicane inclined to the opinion that this curve would be a parabola and not an hyperbola. The Moon-Voyage The ellipse and hyperbola have each two foci, and two corresponding directrixes, and the parabola has one focus and one directrix. The Gutenberg Webster's Unabridged Dictionary Section F, G and H Barbicane was inclined to believe that this curve would be rather a parabola than a hyperbola. From the Earth to the Moon; and, Round the Moon An oval is never mistaken for a circle, nor an hyperbola for an ellipsis. An Enquiry Concerning Human Understanding For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z "Then we have no more hyperbolas, no more parabolas, no more open curves to fear!" The Moon-Voyage In the ellipse this ratio is less than unity, in the parabola equal to unity, and in the hyperbola greater than unity. The Gutenberg Webster's Unabridged Dictionary Section F, G and H One maintained the hyperbola, the other the parabola. From the Earth to the Moon; and, Round the Moon In other positions the projection of the sphere becomes an ellipse, or one of its varieties, the parabola and hyperbola. The Atlantic Monthly, Volume 05, No. 30, April, 1860 For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Algebra and Trigonometry 2015-02-13T00:00:00Z Comets also have their troubles, their perihilions, their hyperbolas and their parabolas. Remarks In other cases they produce hyperbola and parabola. Ancient and Modern Physics "With a certain speed it will assume the parabola, and with a greater the hyperbola." From the Earth to the Moon; and, Round the Moon Some are elliptic, but, for the greater part round: some represent an hemisphere, others a segment, others an hyperbola. Thaumaturgia The center of a hyperbola is the midpoint of both the transverse and conjugate axes, where they intersect. Algebra and Trigonometry 2015-02-13T00:00:00Z Suiting her action to the word, she picked up a red book and sent it in a gentle hyperbola toward the ceiling, where it sank into the lamp beside the first. Tales of the Jazz Age All conic sections, e.g. ellipses, parabolas, and hyperbolas, could be represented by equations. Our Legal Heritage : 600-1776 King Aethelbert - King George III With the stock market approaching the Y2K rollover like a hyperbola approaching its vertical asymptote, all talk of free software or open source as a political phenomenon falls by the wayside. Free as in Freedom: Richard Stallman's Crusade for Free Software Literally, not figuratively, Robert would kiss the place where her foot had trod; but I know that once he rose from such a kiss 'to trace the hyperbola by means of a string.' Robert Falconer Every hyperbola also has two asymptotes that pass through its center. Algebra and Trigonometry 2015-02-13T00:00:00Z For my thought is like a hyperbola that continually widens ascending. The Story of My Heart An Autobiography "Is it then an hyperbola? and are we to be carried on far and away into remote distance, and never, never to return?" Off on a Comet! a Journey through Planetary Space I cannot do more than merely allude to the treatises on Conic Sections and on Maxima and Minima by Apollonius, who is said to have been the first to introduce the words ellipse and hyperbola. History of the Conflict Between Religion and Science Then we will turn our attention to finding standard equations for hyperbolas centered at some point other than the origin. Algebra and Trigonometry 2015-02-13T00:00:00Z As a hyperbola recedes from the center, its branches approach these asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z Finally, we substitute and into the standard form of the equation, The equation of the hyperbola is as shown in Figure 6. Algebra and Trigonometry 2015-02-13T00:00:00Z The central rectangle and asymptotes provide the framework needed to sketch an accurate graph of the hyperbola. Algebra and Trigonometry 2015-02-13T00:00:00Z Next, we plot and label the center, vertices, co-vertices, foci, and asymptotes and draw smooth curves to form the hyperbola, as shown in Figure 9. Algebra and Trigonometry 2015-02-13T00:00:00Z TRY IT #5 Graph the hyperbola given by the standard form of an equation Identify and label the center, vertices, co-vertices, foci, and asymptotes. Algebra and Trigonometry 2015-02-13T00:00:00Z To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle. Algebra and Trigonometry 2015-02-13T00:00:00Z Assume that the center of the hyperbola—indicated by the intersection of dashed perpendicular lines in the figure—is the origin of the coordinate plane. Algebra and Trigonometry 2015-02-13T00:00:00Z First, we find Recall that the length of the transverse axis of a hyperbola is This length is represented by the distance where the sides are closest, which is given as meters. Algebra and Trigonometry 2015-02-13T00:00:00Z What must be true of the foci of a hyperbola? Algebra and Trigonometry 2015-02-13T00:00:00Z If the transverse axis of a hyperbola is vertical, what do we know about the graph? Algebra and Trigonometry 2015-02-13T00:00:00Z By definition of a hyperbola, is constant for any point on the hyperbola. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, determine whether the following equations represent hyperbolas. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, find the equations of the asymptotes for each hyperbola. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, given information about the graph of the hyperbola, find its equation. Algebra and Trigonometry 2015-02-13T00:00:00Z Conversely, an equation for a hyperbola can be found given its key features. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, given the graph of the hyperbola, find its equation. Algebra and Trigonometry 2015-02-13T00:00:00Z For the following exercises, express the equation for the hyperbola as two functions, with as a function of Express as simply as possible. Algebra and Trigonometry 2015-02-13T00:00:00Z Find the equation of the hyperbola and sketch the graph. Algebra and Trigonometry 2015-02-13T00:00:00Z Because so we will graph a hyperbola with a focus at the origin. Algebra and Trigonometry 2015-02-13T00:00:00Z We begin by finding standard equations for hyperbolas centered at the origin. Algebra and Trigonometry 2015-02-13T00:00:00Z |
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