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Hence the asymptote cuts the axis of y at a distance , and that of v at a distance from the origin, and as it is therefore inclined at an angle of 45° to the axis of X, its equation is
y = x +
Therefore y = + (x + a) are the equations to two asymptotes at right angles to each other.
Another asymptote parallel to the axis of y is given by putting x = a. (15) Let the equation to the curve be
23 – 3a x? + a y = - .,
x" – 3bx + 26%.
The denominator equated to o gives x = b, x = 2b; therefore the corresponding ordinates are asymptotes, since for x = b and x = 2b y is infinite.
Whence, expanding and rejecting the terms involving negative powers of Q, we have y = x – 3 (a - b) as the equation to a third asymptote, which is therefore inclined at an angle of 45° to the axis of a.
When the equation cannot be solved with respect to y, we are sometimes able to determine the asymptotes by assuming y = xx, and then by means of the equation expressing X and y in terms of z. If the same value of x which renders 20 and y infinite give a finite value for the intercepts of the tangents, then these determine the position of the asymptotes.
(16) Let a ys – 603 + c^æy = 0 be the equation to the curve: then assuming y = xx, we find
la), and the
Now w and y are both infinite when 8 = intercept of the tangent on the axis of y is
- céry - cz 90 3ay + c*
which when % =
1, and consequently all = oo becomes
and the equation to the asymptote is
(17) If the equation to the curve be
y' - &+ 2bx'y = 0, we find by the same means the equations to two asymptotes to be
(18) Find the asymptotes of the curve
23 – ay (x – b) = 0.
the curve has a rectilinear asymptote in the ordinate at a distance b from the origin. It has also a parabolic asymptote, for we have
and therefore for the asymptote
ay = x2 – bx + b?;
oray - = (x – 56); the equation to a common parabola, the latus rectum of which is a, and the axis of which is parallel to that of y. (19) The curve whose equation is
a'yo – 26'y – ** = 0, has two parabolic asymptotes whose equations are
Their common axis is therefore the axis of y, and their latera recta are equal to a, but they are turned in opposite directions.
It sometimes happens that we obtain an equation for an asymptote with possible coefficients, though for large values of one variable in the equation to the curve, the other variable becomes impossible. This apparent anomaly has been explained by Mr Walton *, by availing himself of the general interpretation which may be given to the symbols in analytical geometry. The impossibility of one of the variables, when certain values are assigned to the other, may be interpreted as signifying that the curve for these values leaves the plane to which it is referred. Now when by assigning an indefinitely large value to the one variable, the other tends to become again possible and to assume the form of the equation to a straight line, as is the case when we find a possible rectilinear asymptote, this indicates that the curve tends to return to the plane of reference, and that at an infinite distance it will coincide with it in a line, the equation to which is that of the asymptote.
(20) As an example of a curve having a possible asymptote to an impossible branch let us take the equation,
* (y - c)? = 6* (a? – ). When x = 0, y = c and is possible, and therefore the axis of y is an asymptote: this is one of the ordinary kind. But if we put the equation under the form
(a” – X*) (y - c) =
24 s it is easily seen that when x = 0, y = C. On the other hand, if > a, y is impossible. Hence the line whose equation is y = c is an asymptote to an impossible branch of the curve; that is to say, a branch of the curve leaves the plane of reference when x = Ła, but tends to return to it again when a = co, coinciding then with the line whose equation is y = C. The form of the curve is given in fig. 27, where the dotted curve represents the impossible branches of the curve lying in a plane at right angles to the plane of the paper.
On the subject of asymptotes to curves, the reader may consult in addition to the work of Stirling before referred to, Newton's Enumeratio Linearum Tertii ordinis, Cramer's Analyse des Lignes Courbes, Chap. viii. and a paper in the Cambridge Mathematical Journal for November, 1843.
* Cambridge Mathematical Journal, Vol. 11. p. 236.
If the curve be expressed by a relation between 7 and 0, then the tangent of the angle (0) between the radius vector
do o and the tangent to the curve is room. The subtangent, which is the portion of a perpendicular to the radius vector
da; and the
at the origin intercepted by the tangent, perpendicular from the origin on the tangent is
If the curve be expressed by a relation between u and o
where u =-, the subtangent and perpendicular are equal to
Asymptotes to spirals are determined by finding what value of @ makes infinite; and if the same value of 0
de . make go? - either finite or equal to zero, a line drawn through the extremity of the subtangent parallel to r is an asymptote to the curve.
Spirals may have asymptotic circles: these are found by the condition that an infinite value of 0 gives a finite value for . Ex. 1. The equation to the spiral of Archimedes is
r = al.