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Handbook of mathematics for engineers and scienteists part 59

Hach triangle has three excircles. The center of an excircle (an excenter) is the point of concurrency of two external angle bisectors and an interior angle bisector. The straight lines connecting the vertices of a triangle with the points at which the respective opposite sides are tangent to the excircles intersect in a single point _V. | 374 Differential Geometry 9.1.1-4. Asymptotes. A straight line is called an asymptote of a curve r if the distance from a point M x y of the curve to this straight line tends to zero as x2 y2 o. The limit position of the tangent to a regular point of the curve is an asymptote the converse assertion is generally not true. For a curve given explicitly as y f x vertical asymptotes are determined as points of discontinuity of the function y f x while horizontal and skew asymptotes have the form y kx b where k lim .f x b lim f x - kx t œ x t œ both limits must be finite. Example 12. Let us find the asymptotes of the curve y x2 x2 1 . Since the limits f x . x3 k lim ----- lim 1 t x x t x x x2 1 3 . x . -x b lim f x - kx lim - - x lim - 0 t x t x x 1 t x x2 17 exist the asymptote is given by the equation y x. To find an asymptote of a parametrically defined curve x x t y y t one should find the values t ti for which x t - o or y t o. If x ti o but y ti c o then the straight line y c is a horizontal asymptote. if y ti o but x ti a o then the straight line x a is a vertical asymptote. if x ti o and y ti o then one should calculate the following two limits k lim and b lim y t - kx t . t ti x t t ti 9.1.1.4 if both limits exist then the curve has the asymptote y kx b. Example 13. Let us find the asymptote of the Folium of Descartes 3at 3at2 x ----- y -------- -œ t œ . t3 1 t3 1 Since x -1 œ and y -1 œ one should use formulas 9.1.1.4 3af2 k lim - lim 1 lim t -1 t -1 x t t -1 ôat t -1 t3 1 3at2 3at 3at t 1 b lim y t - kx t lim ------ ---- lim ------------ -------- t -1Ly V J t -1 t3 1 t3 1 t -1 t 1 t2 -1 1 which imply that the asymptote is given by the equation y -x - a. 9.1. Theory of Curves 375 Suppose that the function F x y in the equation F x y 0 is a polynomial in the variables x and y. We choose the terms of the highest order in F x y . By x y we denote the set of highest-order terms and solve the equation for the variables x and y x Ay y x . The values yi c for which x