If the tangent at P meets the asymptotes in R, R', then CR.
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.
If the asymptotes be perpendicular, or, in other words, the principal axes be equal, the curve is called the rectangular hyperbola.
We may observe that the asymptotes intersect this circle in the same points as the directrices.
This theorem furnishes a ready means of constructing the hyperbola by points when a point on the curve and the two asymptotes are given.
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.
Triangle formed by the two asymptotes and a tangent 118.
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.
Take, now, for two of the tangents the asymptotes of a hyperbola, and let any third tangent cut them in A and B (Fig.
Equation of hyperbola referred to the asymptotes 119.
Given the asymptotes and a finite point of a conic, to construct the conic.
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.
Segments cut off on a chord by hyperbola and its asymptotes 116.
Therefore The asymptotes cut off on each tangent a segment which is bisected by the point of contact.
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.
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.
For the asymptotes as axes of co-ordinates the equation of the hyperbola is xy = const.
The parallelogram PQOQ' formed by the asymptotes and lines parallel to them through P will be half the triangle OHK, and will therefore be constant.
This last equation, of which the left-hand side satisfies the condition for breaking up into two factors, represents the two imaginary circular points at infinity, through which all circles and their asymptotes pass.
Two asymptotes and any two tangents to an hyperbola may be considered as a quadrilateral circumscribed about the hyperbola.
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.
C, D and the asymptotes in E, F, then the line OM which bisects the chord CD is a diameter conjugate to the diameter OK which is parallel to the secant t, so that OK and OM are harmonic with regard to the asymptotes.
If we now take the asymptotes OX and OY as oblique axes of co-ordinates, the lines OQ and QP will be the co-ordinates of P, and will satisfy the equation xy = const.
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.
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.
The hyperbola would have an equation of the form xy = c if referred to its asymptotesas axes, the coordinates being then oblique, unless a = b, in which case the hyperbola is called rectangular.
When a real factor of un is repeated we may have two parallel asymptotes or we may have a "parabolic asymptote.
The asymptotes coincide with the diagonals of the parallelogram formed on any two conjugate diameters.
The portions of a secant comprised between the hyperbola and its asymptotes are equal.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "asymptotes" in a variety of sentences. We hope that you will now be able to make sentences using this word.
