rectilinear-asymptote-from-equation

Rectilinear Asymptote from the Equation

Summary: When the highest member of the curve equation factorises as $P=(ay-bx)M$ with the linear factor occurring once, the curve has a straight-line asymptote $ay-bx=p$ whose constant $p$ is read off from the next member: $p=-Q/M$ evaluated along $y/x=b/a$, i.e. by substituting $b$ for $y$ and $a$ for $x$. Each distinct simple real linear factor of $P$ contributes one such asymptote and a pair of opposite branches; the hyperbola is the simplest specimen.

Sources: chapter7 (§§170-177)

Last updated: 2026-04-28

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Setup

§170. Members of the curve equation $P+Q+R+S+\cdots =0$ are listed by descending degree (see branches-at-infinity). $P$ is degree $n$, $Q$ is degree $n-1$, etc.

§171. Assume $P$ has exactly one real linear factor, $ay-bx$, so

$P=(ay-bx)M$

where $M$ is a homogeneous form of degree $n-1$ with no real linear factors (so $M$ never vanishes at infinity; cf. branches-at-infinity).

If $x$ or $y$ is infinite, then $M={\infty }^{n-1}$. For the equation to hold, $(ay-bx)M$ must be cancelled by the lower members $Q+R+S+\cdots$. Since $Q$ alone can reach ${\infty }^{n-1}$ and $R,S,\dots$ cannot, the only way is for $ay-bx$ itself to stay finite at infinity, while $M$ blows up. That is, the branch heads toward infinity along a direction where $ay-bx$ is bounded — exactly the direction $y/x\to b/a$.

The asymptote constant $p=-Q/M$

§172-173. Set $ay-bx=p$ where $p$ is the (finite) limit value. Then

$pM+Q+R+S+\cdots =0⟹p=\frac{-Q-R-S-\cdots }{M}.$

At infinity, $R/M,S/M,\dots$ vanish (their numerators are of strictly lower degree than $M$), so

$\boxed{p=-\frac{Q}{M}.}$

Both $Q$ and $M$ are homogeneous of degree $n-1$, so $-Q/M$ is a zero-degree function of $x,y$ — its value depends only on the ratio $y/x$. Plug in the asymptotic ratio $y/x=b/a$ (equivalently substitute $y=b$, $x=a$) into $-Q/M$ to get the constant $p$.

§174. The asymptote is therefore the straight line

$ay-bx=p,\text{i.e.}y=\frac{bx+p}{a},\frac{y}{x}\to \frac{b}{a}\text{as }x\to \infty .$

This line is approached by the curve in both directions $x\to +\infty$ and $x\to -\infty$, so the curve has two branches going to infinity in opposite directions, both converging on the same asymptote.

Two distinct linear factors → hyperbola

§175-177. If $P=(ay-bx)(cy-dx)M$ with two distinct real linear factors and $M$ of degree $n-2$ with no real linear factor, the analysis runs case-by-case for each factor:

• For the factor $ay-bx$: substitute $y/x=b/a$ into $-Q/[(cy-dx)M]$ to get a constant $p$; the asymptote is $ay-bx=p$.
• For the factor $cy-dx$: substitute $y/x=d/c$ into $-Q/[(ay-bx)M]$ to get $q$; the asymptote is $cy-dx=q$.

The denominators $bc-ad$ and $da-cb$ are non-zero because the factors are distinct, and $M$ stays finite at neither asymptotic direction, so both constants are well defined (possibly zero, in which case the asymptote passes through the origin).

The curve has two distinct asymptotes and four branches approaching them. For a second-order line ($n=2$, $M$ a non-zero scalar) this is exactly the hyperbola: the highest member $\alpha y^2+\beta xy+\gamma x^2$ has two distinct real linear factors when ${\beta }^2>4\alpha \gamma$ (classification-of-conics, hyperbola).

What about a triple or higher distinct factor structure?

If $P$ has $k$ distinct simple real linear factors, the same argument applies factor-by-factor: each contributes one straight asymptote and one pair of opposite branches, for a total of $k$ asymptotes and $2k$ branches. (§185 confirms this for $k=3$.)

The worked-out cases in chapter 7 — straight, parabolic, curvilinear — are governed by multiplicity, not by count, of the linear factors of $P$:

Factor structure of $P$	Asymptote type
One simple real factor	Straight line (rectilinear-asymptote-from-equation)
Two or more distinct simple factors	Several straight lines
Double factor $(ay-bx{)}^2$	parabolic-asymptote
Triple/higher repeated factor	curvilinear-asymptote

Change of axis to align with the asymptote

A recurring technical move (§§183, 187, 193) — used systematically in parabolic-asymptote and curvilinear-asymptote — is to change coordinates so that one axis lies along the asymptote direction and the other is perpendicular to it. Set $g=\sqrt{a^2+b^2}$ and let

$t=\frac{ax+by}{g},u=\frac{ay-bx}{g}.$

Then $u$ measures perpendicular distance to the line $ay-bx=0$ (so $u=p/g$ on the asymptote) and $t$ runs along the asymptote direction. In these coordinates the asymptotic equation becomes a clean low-order form in $t$ and $u$, suitable for taking the $t\to \infty$ limit.

For the simple-factor case treated here, this just rewrites $ay-bx=p$ as $u=p/g$, a horizontal line in $(t,u)$.

Related pages

• branches-at-infinity
• parabolic-asymptote
• curvilinear-asymptote
• chapter-7-on-the-investigation-of-branches-which-go-to-infinity
• hyperbola
• asymptotes-of-hyperbola
• classification-of-conics