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what will be the equation of common tangent of hyperbola and conjugate hyperbola?

what will be the equation of common tangent of hyperbola and conjugate hyperbola?


1 Answers

askIITians Faculty 747 Points
6 years ago

, one of the conic sections, being that which is made by a plane cutting a cone so, that, entering one side of the cone, and not being parallel to the opposite side, it may cut the circular base when the opposite side is ever so far produced below the vertex, or shall cut the opposite side of the cone produced above the vertex, or shall make a greater angle with the base than the opposite side of the cone makes; all these three circumstances amounting to the same thing, but in other words.

1. Thus, the figure DAE is an Hyperbola, made by a plane entering the side VQ of a cone PVQ at A, and either cutting the bafe PEQ when the plane is not parallel to VP, and this is ever so far produced; or when the angle ARQ is greater than the angle VPQ; or when the plane cuts the opposite side in B above the vertex.

2. By the Hyperbola is sometimes meant the whole plane of the section, and sometimes only the curve line of the section.

3. Hence, the cutting plane meets the opposite cone in B, and there forms another Hyperbola d B e, equal to the former one, and having the same transverse axis AB; and the same vertices A and B. Also the two are called Opposite Hyperbolas.

4. The centre C is the middle point of the tranverse axis.

5. The semi-conjugate axis is CL, a mean proportional between CI and CK, the distances to the sides of the opposite cone, when CI is drawn parallel to the diameter PQ of the base of the cone. Or the whole conjugate axis is a mean proportional between AF and BH, which are drawn parallel to the base of the cone.

6. If DAE and FBG be two opposite Hyperbolas, having the same transverse and conjugate axes AB and a b, perpendicularly bisecting each other; and if d a e and f b g be two other opposite Hyperbolas, having the same axes with the two former, but in the contrary order, viz, having a b for their first or transverse axis, and AB for their second or conjugate axis: then any two adjacent curves are called Conjugate Hyperbolas, and the whole figure formed by all the four curves, the Figure of the Conjugate Hyperbolas. And if the rectangle HIKL be inscribed within the four conjugate Hyperbolas, touching the vertices A, B, a, b, and having their sides parallel and equal to the two axes; and if then the two diagonals HCK, ICL, of the parallelogram be drawn, these diagonals are the asymptotes of the curves, being lines that continually approach nearer and nearer to the curves, without meeting them, except at an infinite distance, where each asymptote and the two adjacent sides of the two conjugate Hyperbolas may be supposed all to meet; the asymptote being there a common tangent to them both, viz, at that infinite distance.

7. Hence the four Hyperbolas, meeting and running into each other at the infinite distance, may be cons<*>dered as the four parts of one entire curve, having the same axes, tangents, and other properties.

8. A Diameter in general, is any line, as MN, drawn through the centre C, and meeting, or termi- nated by the opposite legs of the opposite Hyperbolas. And if parallel to this diameter there be drawn two tangents, at m and n, to the opposite legs of the other two opposite Hyperbolas, the line mCn joining the points of contact, is the conjugate diameter to MN, and the two mutually conjugates to each other. Or, if to the points M or N there be drawn a tangent, and through the centre C the line mn parallel to it, that line will be the conjugate to MN. The points where each of these meet the curves, as M, N, m, n, are the vertices of the diameters; and the tangents to the curves at the two vertices of any diameter, are parallel to each other, and also to the other or conjugate diameter.

9. Moreover, if those tangents to the four Hyperbolas, at the vertices of two conjugate diameters, be produced till they meet, they will form a parallelogram OPQR; and the diagonals OQ and PR of the parallelogram will be the asymptotes of the curves; which therefore pass through the opposite angles of all the parallelograms so inscribed between the curves. Also it is a property of these parallelograms, that they are all equal to each other, and therefore equal to the rectangle of the two axes; as will be farther noticed below. Farther, if these diagonals or asymptotes make a right angle between them, or if the inscribed parallelogram be a square, or if the two axes be equal to each other, then the Hyperbola is called a right-angled or an equilateral one.

10. An Ordinate to any diameter, is a line drawn parallel to its conjugate, or to the tangent at its vertex, and terminated by the diameter produced and the curve. So MS and TN are ordinates to the axis AB; also AD and BG are ordinates to the diameter MN (last fig. but one). Hence the ordinates to the axis are perpendicular to it; but ordinates to the other diameters are oblique to them.

11. An absciss is a part of any diameter, contained between its vertex and an ordinate to it; and every ordinate has two abscisses: as AT and BT, or MV and NV.

12. The Parameter of any diameter, is a third proportional to the diameter and its conjugate.—The Parameter of the axis is also equal to the line AG or Bg (fig. 1), if FG be drawn to make the angle AFG = the angle BAV, or the line Hg to make the angle BHg = the angle ABV.

13. The Focus is the point in the axis where the ordinate is equal to half the parameter of the axis; as S and T (fig. 2) if MS and TN be half the parámeter, or the 3d proportional to CA and Ca. Hence there are two Foci, one on each side the vertex, or one for each of the opposite Hyperbolas. These two points in the axis are called Foci, or burning points, because it is found by opticians that rays of light issuing from one of them, and falling upon the curve of the Hyperbola, are reflected into lines that verge towards the other point or Focus. To describé an Hyperbola, in various ways.

14. (1st Way by points.)—In the transverse axis AF produced, take the foci F and f, by making CF and Cf = Aa or Ba, assume any point I: Then with the radii AI, BI, and centres F, s, describe arcs intersecting in E, which will give four points in the curves. In like manner, assuming other points I, as many other points will be found in the curve. Then, with a steady hand, draw the curve line through all the points of intersection E.—In the same manner are to be constructed the other pair of opposite Hyperbolas, using the axis ab instead of AB.

15. (2d Way by points, for a Right-angled Hyperbola only.)—On, the axis produced if necessary, take any point I, through which draw a perpendicular line, upon which set off IM and IN equal to the distance Ia or Ib from I to the extremities of the other axis; and M and N will be points in the curve.

16. (3d Way by points, to describe the curve through a given point.)—CG and CH being the asymptotes, and P the given point of the curve; through the point P draw any line GPH between the asymptotes, upon which take GI = PH, so shall I be another point of the curve. And in this manner may any number of points be found, drawing as many lines through the given point P.

17. (4th Way by a continued Motion.)—If one end of a long ruler fMO be fastened at the point f, by a pin on a plane, so as to turn freely about that point as a centre. Then take a thread FMO, shorter than the ruler, and fix one end of it in F, and the other to the end O of the ruler. Then if the ruler fMO be turned about the fixed point f, at the same time keeping the thread OMF always tight, and its part MO close to the side of the ruler, by means of the pin M; the curve line AX described by the motion of the pin M is one part of an Hyperbola. And if the ruler be turned, and move on the other side of the fixed point F, the other part AZ of the same Hyperbola may be described after the same manner.—But if the end of the ruler be sixed in F, and that of the thread in f, the opposite Hyperbola xaz may be described.

18. (5th Way, by a continued Motion.)—Let C and F be the two foci, and E and K the two vertices of the Hyperbola. (See the last fig. above.) Take three rulers CD, DG, GF, so that CD = GF = EK, and DG = CF; the rulers CD and GF being of an indefinite length beyond C and G, and having slits in them for a pin to move in; and the rulers having holes in them at C and F, to fasten them to the foci C and F by means of pins, and at the points D and G they are to be joined by the ruler DG. Then, if a pin be put in the slits, viz, the common intersection of the rulers CD and GF, and moved along, causing the two rulers GF, CD, to turn about the foci C and F, that pin will describe the portion Ee of an Hyperbola.—The foregoing are a few among various ways given by several authors. Some of the chief Properties os the Hyperbola.

19. (1st) The squares of the ordinates, of any diameter, are to each other, as the rectangles of their abscisses; i. e. .

20. As the square of any diameter, is to the square of its conjugate; so is the rectangle of two abscisses, to the square of their ordinate. That is, .

Or, because the rectangle AD . BD is = the difference of the squares CD2 - CB2, the same property is, , Or ,

That is , where p is the parameter of the diameter AB, or the 3d proportional ab2/(AB).

And hence is deduced the common equation of the Hyperbola, by which its general nature is expressed. Thus, putting d = the semidiameter CA or CB, c = its semiconjugate Ca or Cb, p = its parameter or 2d2/c, x = the absciss BD from the vertex, y = the ordinate DE, and v = the absciss CD from the centre: Then is,or,or,or; so that <*> any of which equations or proportions express the nature of the curve. And hence arises the name Hyperbola, signifying to exceed, because the ratio of d2 to c2, or of d to p, exceeds that of 2dx to y2; that ratio being equal in the parabola, and defective in the ellipse, from which circumstances also these take their names.

21. The distance between the centre and the focus, is equal to the distance between the extremities of the transverse and conjugate axes. That is, CF = Aa or Ab, where F is the focus.

22. The conjugate semi-axis is a mean proportional between the distances of the focus from both vertices of the transverse. That is, Ca is a mean between AF and BF, or , or .

23. The difference of two lines drawn from the foci, to meet in any point of the curve, is equal to the transverse axis. That is, fE - FE = AB, where F and f are the two foci.

24. All the parallelograms inscribed between the four conjugate Hyperbolas are equal to one another, and each equal to the rectangle of the two axes. That is, the parallelogram OPQR = AB . ab (fig. to art. 9).

25. The difference of the squares of every pair of conjugate diameters, is equal to the same constant quantity, viz, the difference of the squares of the two axes. That is, , (fig. to art. 6)<*> where MN and mn are any two conjugate diameters.

26. The rectangles of the parts of two parallel lines, terminated by the curve, are to one another, as the rectangles of the parts of any other two parallel lines, any where cutting the former. Or the rectangles of the parts of two intersecting lines, are as the squares of their parallel diameters, or squares of their parallel tangents.

27. All the rectangles are equal which are made of the segments of any parallel lines, cut by the curve, and limited by the asymptotes, and each equal to the square of their parallel diameter. That is, HE . EK or or CP2.

28. All the parallelograms are equal, which are formed between the asymptotes and curve, by lines parallel to the asymptotes. That is, the paral. CGEK = CPBQ.—Hence is obtained another method of expressing the nature of the curve by an equation, involving the absciss taken on one asymptote, and ordinate parallel to the other asymptote. Thus, if x = CK, y = KE, a = CQ, and b = BQ the ordinate at the vertex B of the curve; then, by the property in this article, ab = xy, or ; that is, the rectangle of the absciss and ordinate is every where of the same magnitude, or any ordinate is reciprocally as its absciss.

29. If the abscisses CQ, CK, CL, &c, taken on the one asymptote, be in geometrical progression increasing; then shall the ordinates QB, KE, LM, &c, parallel to the other asymptote, be a like geometrical progression in the same ratio, but decreasing; and all the rectangles are equal, under every absciss and its ordinate, viz, , &c.

30. The abscisses CQ, CK, CL, &c, being taken in geometrical progression; the spaces or asymptotie areas BQKE, EKLM, &c, will be all equal; or, the spaces BQKE, BQLM, &c, will be in arithmetical progression; and therefore these spaces are the hyperbolic logarithms of those abscisses.

These, and many other curious properties of the Hyperbola, may be seen demonstrated in my Treatise on Conic Sections, and several others. See also Conic Sections.

Acute Hyperbola, one whose asymptotes make an acute angle.

Ambigenal Hyperbola, is that which has one of its infinite legs falling within an angle formed by the asymptotes, and the other falling without that angle. This is one of Newton's triple Hyperbolas of the 2d order. See his Enumeratio Lin. tert. Ord. See also Ambigenal.

Common, or Conic Hyperbola, is that which arises from the section of a cone by a plane; called also the Apollonian Hyperbola, being that kind treated on by the first and chief author Apollonius.

Conjugate Hyperbolas, are those formed or lying together, and having the same axes, but in a contrary order, viz, the transverse of each equal the conjugate of the other; as the two Conjugate Hyperbolas Pee and EEE in the last figure but one.

Equilateral, or Rectanglar Hyperbola, is that whose two axes are equal to each other, or whose asymptotes make a right angle.—Hence, the property or equation of the equilateral Hyperbola, is , where a is the axis, x the absciss, and y its ordinate; which is similar to the equation of the circle, viz, , differing only in the sign of the second term, and where a is the diameter of the circle.

Infinite Hyperbolas, or Hyperbolas of the higher kinds, are expressed or defined by general equations similar to that of the conic or common Hyperbola, but having general exponents, instead of the particular numeral ones, but so as that the sum of those on one side of the equation, is equal to the sum of those on the other side. Such as, , where x and y are the absciss and ordinate to the axis or diameter of the curve; or , where the absciss x is taken on one asymptote, and the ordinate y parallel to the other.

As the Hyperbola of the first kind, or order, viz the conic Hyperbola, has two asymptotes; that of the 2d kind or order has three; that of the 3d kind, four; and so on.

Obtuse Hyperbola, is that whose asymptotes form an obtuse angle.

Rectangular Hyperbola, the same as Equilateral Hyperbola.

Hyperbolic Arc, is the arc of an Hyperbola.

Put a = CA the semitransverse axe, c = Ca the semiconjugate, y = an ordinate PQ to the axe drawn from the end Q of the arc AQ, beginning at the vertex A: then putting , &c; then is the length of the arc AQ expressed by &c; or by , nearly; where t is the whole transverse axe 2CA, c = 2Ca the conjugate, x = AP the absciss, and y = PQ the ordinate.

These and other rules may be seen demonstrated in my Mensuration, p. 408, &c, 2d edit.

Hyperbolic Area, or Space, the area or space included by the Hyperbolic curve and other lines.

Putting a = CA the semitransverse, c = Ca the semiconjugate, y = PQ the ordinate, and v = CP its distance from the centre; then is the area ; sector ; area ; or nearly.

Let CT and CE be the two asymptotes, and the ordinates DA, EF parallel to the other asymptote CT; then the asymptotic space ADEF or sector CAF is or or &c; and this last series was first given by Mercator in his Logarithmotechnia.

See my Mensuration, p. 413, &c, 2d edit.

Generally, if be an equation expressing an Hyperbola of any order; then its asymptotic area will be ; which space therefore is always quadrable, in all the orders of Hyperbolas, except the first or common Hyperbola only, in which m and n being each 1, the denominator n - m becomes 0 or nothing.

Hyperbolic Conoid, a solid formed by the revolution of an Hyperbola about its axis, otherwise called an Hyperboloid. To find the Solid Content of an Hyperboloid.

Let AC be the semitransverse of the generating Hyperbola, and AH the height of the solid; then as 2AC + AH is to 3AC + AH, so is the cone of the same base and altitude, to the content of the Conoid. To find the Curve Surface of an Hyperboloid.

Let AC be the semitransverse, and AB perpendicular to it, and equal to the semiconjugate of ADE the generating Hyperbola, or section through the axis of the solid. Join CB; make CF = CA, and on CA let fall the perpendicular FG; then with the semitransverse CG, and semiconjugate GH = AB, describe the Hyperbola GIK; then as the diameter of a circle is to its circumference, so is the Hyperbolic frustum ILAMK to the curve surface of the Conoid generated by DAE. See my Mensur. p. 429, &c, 2d edit.

Hyperbolic Cylindroid, a solid formed by the revolution of an Hyperbola about its conjugate axis, or line through the centre perpendicular to the transverse axis. This solid is treated of in the Philos. Trans. by Sir Christopher Wren, where he shews some of its properties, and applies it to the grinding of Hyperbolical Glasses; affirming that they must be formed this way, or not at all. See Philos. Trans. vol. 4, pa. 961.

Hyperbolic Leg, of a curve, is that having an asymptote, or tangent at an infinite distance.—Newton reduces all curves, both of the first and higher kinds, into Hyperbolic and parabolic legs, i. e. such as have asymptotes, and such as have not, or such as have tangents at an infinite distance, and such as have not.

Hyperbolic Line, is used by some authors for what is more commonly called the Hyperbola itself, being the curve line of that figure; in which sense the surface terminated by it is called the Hyperbola.

Hyperbolic Logarithm, a logarithm so called as being similar to the asymptotic spaces of the Hyperbola. The Hyperbolic logarithm of a number, is to the common logarithm, as 2.3025850929940457 to 1, or as 1 to .4342944819032518. The first invented logarithms, by Napier, are of the Hyperbolic kind; and so are Kepler's. See Logarithm.

Hyperbolic Mirror, is one ground into that shape.

Hyperbolic Space, that contained by the curve of the Hyperbola, and certain other lines. See HYPERBOLIC Area.

HYPERBOLICUM Acutum, a solid made by the revolution of the infinite area or space contained between the curve of the Hyperbola, and its asymptote. This produces a solid, which though infinitely long and generated by an infinite area, is nevertheless equal to a finite solid body; as is demonstrated by Torricelli, who gave it this name.

HYPERBOLIFORM Figures, are such curves as approach, in their properties, to the nature of the Hyperbola; called also Hyperboloides.

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