The diameter which being produced passes through the foci of the hyperbola is the transverse axis.
The ellipse and hyperbolahave each two foci, and two corresponding directrixes, and the parabola has one focus and one directrix.
Defn: A curve in the form of the figure 8, with both parts symmetrical, generated by the point in which a tangent to an equilateral hyperbolameets the perpendicular on it drawn from the center.
It is an ellipse when the curvature is synclastic, and an hyperbola when the curvature is anticlastic.
Thus rays from A in the ellipse are reflected to B; rays from A in the hyperbola are reflected toward L and M away from B.
Hyperboloid of revolution, an hyperboloid described by an hyperbola revolving about one of its axes.
In the ellipse this ratio is less than unity, in the parabola equal to unity, and in the hyperbola greater than unity.
In the ellipse the sum of the focal distances of any point on the curve is constant, and in the hyperbola the difference between the focal distances is constant.
This theorem furnishes a ready means of constructing the hyperbola by points when a point on the curve and the two asymptotes are given.
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.
Show in this way that if a hyperbola and its asymptotes be cut by a transversal, the segments intercepted by the curve and by the asymptotes respectively have the same middle point.
Apply the theorem of problem 9 to the case of a hyperbola where the two tangents are the asymptotes.
Segments cut off on a chord by hyperbola and its asymptotes 116.
Therefore A’A" = B’B" and we have the theorem The segments cut off on any chord between the hyperbola and its asymptotes are equal.
This identifies the curve with the hyperbola as defined and discussed in works on analytic geometry.
The ellipse and the hyperbola have two foci and two directrices.
Even the hyperboloid of two sheets, obtained by revolving the hyperbola about its major axis, was known to them, but probably not the hyperboloid of one sheet, which results from revolving a hyperbola about the other axis.
In the ellipse the vertex is nearer to the focus than it is to the directrix, for the same reason, and in the hyperbola it is farther from the focus than it is from the directrix.
Then, since A, P, B, and the point at infinity on AB are four harmonic points, we have the theorem Conjugate diameters of the hyperbola are harmonic conjugates with respect to the asymptotes.
Equation of hyperbola referred to the asymptotes 119.
The ratio of the distance between the center and the focus of an ellipse orhyperbola to its semi-transverse axis.
A curve in the form of the figure 8, with both parts symmetrical, generated by the point in which a tangent to an equilateral hyperbolameets the perpendicular on it drawn from the center.
The diameter which being produced passes through the foci of the hyperbola is the transverse axis.
The curve is a conic section--an hyperbola in these regions.
Referred to the asymptotes as axes the general equation becomes xy = k^2; obviously the axes are oblique in the general hyperbola and rectangular in the rectangular hyperbola.
The hyperbola which has for its transverse and conjugate axes the transverse and conjugate axes of another hyperbola is said to be the conjugate hyperbola.
The geometry of the rectangular hyperbola is simplified by the fact that its principal axes are equal.
Numerical exercises relative to the area of equilateral hyperbola or the calculation of a logarithm.
Form of the equation of the hyperbola referred to its asymptotes.
The equation to the tangent, and the normal to the circle, parabola, ellipse, and hyperbolareferred to rectangular axes, the sections of a right cone made by a plane.
The portions of a secant comprised between thehyperbola and its asymptotes are equal.
When the conjugate axis of the hyperbola increases without limit, the loops of the nodoid are crowded on one another, and each becomes more nearly a ring of circular section, without, however, ever reaching this form.
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.
When the conic is a hyperbola the meridian line is in the form of a looped curve (fig.
Fermat described his method of integration as a logarithmic method, and thus it is clear that the relation between the quadrature of the hyperbola and logarithms was understood although it was not expressed analytically.
An oval is never mistaken for a circle, nor an hyperbola for an ellipsis.
Gregory St. Vincent is the greatest of circle-squarers, and his investigations led him into many truths: he found the property of the arc of the hyperbola which led to Napier's logarithms being called hyperbolic.
If we describe on a diameter AB of an ellipse or hyperbola a circle concentric to the conic, it will cut the latter in A and B (fig.
This gives-- An ellipse as well as an hyperbola has one pair of axes.
It follows in an ellipse the ratio between the distance of a point from the focus to that from the directrix is less than unity, in the parabola it equals unity, and in the hyperbola it is greater than unity.
Hence also-- From the centre of an hyperbola two tangents can be drawn to the curve which have their point of contact at infinity.
For the asymptotes as axes of co-ordinates the equation of the hyperbola is xy = const.
The hyperbolawould 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.
The axes thus chosen for the ellipse and hyperbola are called the principal axes.
As the axes are conjugate diameters at right angles to one another, it follows (S 23)-- The axes of an hyperbola bisect the angles between the asymptotes.
Two asymptotes and any two tangents to an hyperbola may be considered as a quadrilateral circumscribed about the hyperbola.
Hence-- Any two conjugate diameters of anhyperbola are harmonic conjugates with regard to the asymptotes.
Similarly, in an hyperbola a vertex is nearer to the directrix than to the focus.
It follows that DF = EC, and ED = CF; or On any secant of an hyperbola the segments between the curve and the asymptotes are equal.
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.
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.
The following analysis shows that with the aid of an hyperbola any arc, and therefore any angle, may be trisected.
This is the equation of an hyperbola whose center is on the axis of abscisses.
What I particularly like in your definition of the hyperbola (I was going to say hyperblague) is that it is still more obscure than the word you pretend to define.
To assimilate the hyperbola to the ellipse was rather to contradict this evidence.
What is true of a real quantity, said Poncelet, should be true of an imaginary quantity; what is true of the hyperbola whose asymptotes are real, should then be true of the ellipse whose asymptotes are imaginary.
Thus the curve of the first order or right line consists of one branch; but in curves of the second order, or conics, the ellipse and the parabola consist each of one branch, the hyperbola of two branches.
The epithets hyperbolic and parabolic are of course derived from the conic hyperbola and parabola respectively.
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.
This principle is thus stated by Newton:--"In parabola velocitas ubiquo equalis est velocitati corporis revolventis in circulo ad dimidiam distantiam; in ellipsi minor est in hyperbola major.
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.
In the hyperbola we have the mathematical demonstration of the error of an axiom.
Again: it is mathematically demonstrable, that a straight line, the asymptote of a hyperbola, may eternally approach the curve of the hyperbola and never meet it.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "hyperbola" in a variety of sentences. We hope that you will now be able to make sentences using this word.
