CHAPTER VIII.

ASYMPTOTES.

208. DEF. If a straight line cut a curve in two points at an infinite distance from the origin and yet is not itself wholly at infinity, it is called an asymptote to the

curve.

209. Equations of the Asymptotes.

Let the equation of any curve of the nth degree be arranged in homogeneous sets of terms and expressed as

To find where this curve is cut by any straight line

whose equation is

y=μx+B......

B

У
X

(B)

substitute μ+ for in equation (A), and the resulting

gives the abscissae of the points of intersection.

Applying Taylor's Theorem to expand each of these

functional forms, equation (c) may be written

x*$n(μ)+xn-1 | Bp'n(μ)+an-2 B2

+n-1(u)

2! $"n(u)+...

=0. (D)

|+ßp'n-1(u)

|+Pn-2(μ)

This is an equation of the nth degree, proving that a straight line will in general intersect a curve of the nth degree in n points real or imaginary.

The straight line y=ux+ß is at our choice, and therefore the two constants μ and B may be chosen, so as to satisfy any pair of consistent equations.

Suppose we

(E)

(F)

The two highest powers of a now disappear from equation (D), and that equation has therefore two infinite roots.

If, then, M, M., μn be the n values of μ deduced from equation (E) (which is of the nth degree in μ), the corresponding values of ß will in general be given by

Hence, in order to find the asymptotes of any given curve, we may either substitute ux+B for y in the equation of the curve, and then by equating the coefficients of

μ

the two highest powers of x to zero find μ and B. Or we may assume the result of the preceding article, which may be enunciated in the following practical way :-In the highest degree terms put x=1 and y=μ [the result of this is to form pn(μ)] and equate to zero. Hence find u. Form pa-1(u) in a similar way from the terms of degree n-1, and differentiate pn(u), then the values of ẞ are found by substituting the several values of μ in the formula

B=

ФП-1(м)
D'n (μ)

Ex. Find the asymptotes of the cubic

Here

therefore

giving Again,

and

therefore

Hence if

if

and if

2x3-x2y-2xy2+y3+2x2+xy-y2+x+y+1=0.
$z(μ) = μ3 – 2μ2 — μ+2=0;

211. Number of Asymptotes to a Curve of the nth Degree. It is clear that since (u)=0 is in general of the nth degree in u, and ßp'n(μ)+Pn-1(μ)=0 is of the first degree in ß, that n values of μ, and no more, can be found from the first equation, while the n corresponding values of B can be found from the second. Hence n asymptotes, real or imaginary, can be found for a curve of the nth degree.

212. If the degree of an equation be odd it is proved in Theory of Equations that there must be one real root at least. Hence any curve of an odd degree must have at least one real asymptote, and therefore must extend to infinity. No curve therefore of an odd degree can be closed. Neither can a curve of odd degree have an even number of real asymptotes, or a curve of even degree an odd number.

213. If, however, the term yn be missing from the terms of the nth degree in the equation of the curve, the term " will also be missing from the equation (u)=0, and there will therefore be an apparent loss of degree in this equation. It is clear, however, that in this case, since the coefficient of u" is zero, one root of the equation Pn(u)=0 is infinite, and therefore the corresponding asymptote is at right angles to the axis of x; i.e., parallel to that of y. This leads us to the special consideration of such asymptotes as may be parallel to either of the axes of co-ordinates.

214. Asymptotes Parallel to the Axes.

Let the curve arranged as in equation (A), Art. 209, be

the coefficients of the two highest powers of x in equation (B′) vanish, and therefore two of its roots are infinite. Hence the straight line a1y+b1=0 is an asymptote.

In the same way, if an=0, ɑn-1x+bn-1=0 is an asymptote.

1

Again, if a=0, a1=0, b1 =0, and if y be so chosen that ay2+b2y+c2=0,

three roots of equation (B') become infinite, and the lines represented by

represent a pair of asymptotes, real or imaginary, parallel to the axis of y.

Hence the rule to find those asymptotes which are parallel to the axes is, "equate to zero the coefficients of the highest powers of x and y.”

Ex. Find the asymptotes of the curve

x2y2x2y-xy2+x+y+1=0.

Here the coefficient of x2 is y2-y and the coefficient of y2 is x2 — x. Hence x=0, x=1, y=0, and y=1 are asymptotes. Also, since the curve is one of the fourth degree, we have thus obtained all the asymptotes.