CHAPTER II. ASYMPTOTES. Definition of an Asymptote. Method of finding Asymptotes. 109. AN asymptote is a tangent line to a curve, such that, although the distance between the origin and the point of contact is infinite, the perpendicular distance of the origin from the line is finite. Ex. 1. Take for instance the curve where a is supposed to be a number greater than unity: then, by the formula du y' du = dx dy dx du we have, for the equation to the tangent at any point, log a . a* . x' - y' = log a . a* . x Suppose now that x = = x log a. a* y απ. ∞: then a 0, and 2 x.a* = az 1 = 2 = ∞, = 0. log a α Thus the equation to the tangent becomes y' = 0, the equation to an asymptote coincident with the axis of coordinates of the point of contact being Then the equation to the tangent will be (2ax + 3x2) x' - 3y3y' = 2ax2 + 3 (x3 – y3) = -- ax3, Suppose that x=∞; then from the equation to the curve we have a y3 y x = 1: From the preceding observations it is plain that the following may be regarded as a general method of finding asymptotes. Assume x = ∞, or y ∞, and then ascertain whether in each case the value of either of the intercepts x, y, of the tangent is finite if one or both of the quantities x, y, be finite, they will correspond to the existence of an asymptote, the position of which they will define. Asymptotes of Algebraic Curves. 110. The method of determining the asymptotes of curves, which has been developed in the preceding Article, although always applicable, is not however so convenient in the case of algebraic curves as the one which we shall now propose. The equation to the tangent at any point (x, y) of a curve may be written in the form λ, u, r, being all regarded as variable. Then the equation to the tangent becomes Suppose now that, to render either x or y infinite, we equate r to zero, and let the corresponding values of λ, μ, dr dr dλ du dr ; then the equation becomes αλ 'du be denoted by (A), (u), -- (X) y' – (μ) x' = (X) ( dire dr which will be the equation to an asymptote, provided that the ratios between (X), (μ), and (x) ( αμ dr ()(), having been dr evaluated, it is reduced to the form of the equation to a line passing within a finite distance from the origin. Remarks on the Equation of the preceding Article. 111. Before proceeding to apply the equation of the preceding Article to particular examples, it is important to shew that the value of αμ dr depends solely upon the product of (λ) or (μ), and a function of the ratio between (λ) and (μ); and that the ratio between (λ) and (u) may be always determined. The equation to the curve, when x, y, respectively, and when negative powers of r have been eradicated, may be written under the form n being the degree of the terms of highest dimensions in X' Differentiating with regard to r we see that, r being after depends only upon the product of (μ) and a function of the ratio of λ to μ. Now the ratio between (A) and (u), putting r0, is discoverable from the homogeneous equation $ {(λ), (μ)} = 0, (λ, μ) representing the terms in v of highest dimensions in λ and μ. Hence the equation for finding asymptotes may, for each asymptote of the curve, be reduced to the equation for a definite straight line. |
