 We may observe that the asymptotes intersect this circle in the same points as the directrices. An important property is: the difference of the focal distances of any point on the curve equals the transverse axis. The tangent at any point bisects the...  The Cone and Its Sections Treated Geometrically - Page 9by S. A. Renshaw - 1875 - 148 pagesFull view - About this book
 | Henry Parr Hamilton - Geometry, Analytic - 1826 - 354 pages
...corresponding abscissa. 143. COR. In the ellipse, SP + HP = Qa, and in the hyperbola, HP — SP = 2 a, that is, in the ellipse, the sum, and in the hyperbola, the difference, of the focal distances, is equal to the major axis. 144. Observation. If the curve, instead of being referred to its principal... | |
 | 1845 - 356 pages
...extremities of any diameters of an ellipse or hyperbola, and a third tangent be drawn to cut these ; in the ellipse, the sum, and in the hyperbola the difference of the angles subtended at the foci by the intercepted portion of the tangent, is equal to two right angles.... | |
 | Mathematics - 1856 - 358 pages
...side of the triangle, is equal to the supplement of the angle * contained by the other two sides. 2. In the ellipse, the sum, and in the hyperbola the difference of tlie angles, subtended at the foci, by any side of the triangle, is equal to the supplement of the... | |
 | W. P. Turnbull - Geometry, Analytic - 1867 - 276 pages
.../ ., . A * PM * BG * the equation ^-1 2 = 1 asserts that ^.AM** AC*' Also SP = e.ME=e(CM-CE)=ex-a, The difference of the focal distances of any point on the curve is the constant 2a. 146. There is no restriction as to the relative magnitude of a and 6. For, as the... | |
 | William Peveril Turnbull - Geometry, Analytic - 1867 - 298 pages
...asserts that . -, , .,,,= -TT- . ua 6s AM. A'M AC* Also SP = e.ME=e(CM-CE) = ex-a, S'P=e.ME'=e(CM+CE') = The difference of the focal distances of any point on the curve is the constant 2a. 146. There is no restriction as to the relative magnitude of a arid b. For, as the... | |
 | Thomas Grenfell Vyvyan - Geometry, Analytic - 1867 - 294 pages
...directrices, PN perpendicular to &S", and join SP.S'P. Then = e.PX=e.LN, .:S'P-SP=eL'L. S'A! U LASN Hence in the hyperbola the difference of the focal distances of any point is constant. When P coincides with A, S'A - SA=AA', .:S'P-SP = AA'; TT r,T CA a Hence CL = — = -... | |
 | Dublin city, univ - 1871 - 366 pages
...curve consists of one closed oval whose points correspond symmetrically in pairs, and that the sum of the focal distances of any point on the curve is equal to the axis major. i1. ACB is a spherical triangle, having the angle at fright, the vertex 6' is given in position,... | |
 | George Shoobridge Carr - Mathematics - 1880
...PBOOF-BYÍ1151) c-A8_A'8_b (A'S±A8) _ CS GA ,(».„ JJy (UÖ1J, e- AX~AX-i (AX±AX) - OA or ОГ 1163 In the ellipse the sum, and in the hyperbola the difference, of the focal distances of P is equal to the major axis, or PS'±PS = AA'. PEOOF. — With the same figures we have, in the ellipse,... | |
 | Samuel Earnshaw - Differential equations, Partial - 1881 - 602 pages
...NX~~NX' is constant, and therefore SP~ SP' is constant and equal to SA ~ S'A or AA' ; that is to say, the difference of the focal distances of any point on the curve is equal to the transverse axis. Corollary 1. Since 8A :AX= 8 A' : A'X = \ (8A i 8 A') : \ (AX+ A'X}; and since the... | |
 | Charles Taylor - Mathematics - 1881 - 512 pages
...NX' is constant, and therefore /SP~ SP1 is constant and equal to SA ~ S'A or A A'; that is to say, the difference of the focal distances of any point on the curve is equal to the transverse axis. Corollary 1. Since SA :AX= SA' : A'X = \ (SA t SA'} : J (AX + A'X] ; and since the... | |
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