asymptotes-of-hyperbola

Asymptotes of the Hyperbola

Summary: As the abscissa of a hyperbola grows without bound, the tangent direction approaches a fixed line through the centre with slope $\pm b/a$. These two lines — never meeting the curve, but coinciding with it at infinity — are the asymptotes, the projective analogue of the parabola’s directrix. Their angle has tangent $2ab/(a^2-b^2)$, becoming a right angle when $a=b$ (the equilateral hyperbola). Taking an asymptote as axis and the parallel to the other as ordinate, the hyperbola has the simple equation $xy=(a^2+b^2)/4$. In this frame: any chord parallel to $Gh$ has midpoint on the curve where the tangent meets it, the rectangle $QM\cdot QN$ between curve and asymptote is constant, and the rectangle $Mm\cdot Mn=b^2$ between curve and asymptote — Apollonius’s principal asymptotic property — drops out as a one-line algebraic identity.

Sources: chapter6 §§158-165; figures 33-34 (in figures33-34)

Last updated: 2026-04-26

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Tangent at infinity (§158)

In the hyperbola equation $y^2=(b^2/a^2)(x^2-a^2)$, the tangent at $M$ meets the principal axis at $T$ with $CT=a^2/x$. As $x\to \infty$, $CT\to 0$ — the tangent passes through the centre $C$. Its slope is

$\tan \angle PTM=\frac{PM}{PT}=\frac{b^{2}x}{a^{2}y},y=\frac{b\sqrt{x^2-a^2}}{a}\to \frac{bx}{a}(x\to \infty ),$

so $\tan \angle ACD=b/a$ in the limit. Erect the perpendicular $AD$ at $A$ with length $b$; the line $CD$ extended in both directions never touches the curve, but the curve approaches it as closely as we please. The line on the opposite side $CK$ behaves the same way for the branches $Bk,Bi$. Such a line is called an asymptote (source: chapter6, §158). The hyperbola has two asymptotes, $ICk$ and $KCi$, crossing at $C$.

Equilateral hyperbola (§159)

The two asymptotes make angle $\angle ACD=\angle ACd$ with the axis, both with tangent $b/a$. The total angle $\angle DCd$ between them has tangent $\frac{2ab}{a^2-b^2}$. When $b=a$ this denominator vanishes, so $\angle DCd=90°$ — the asymptotes are perpendicular. The hyperbola is then called equilateral. In that case $CD=Cd=\sqrt{a^2+b^2}=CG$, so the foci coincide with the asymptote-perpendicular feet from the centre, and the perpendiculars $GH$ from a focus to the asymptotes have $CH=a$ and $GH=b$.

Constant rectangle on chord-asymptote intervals (§160)

Take an arbitrary chord $MPN$ parallel to the conjugate axis ($y$-axis), extended to meet the asymptotes at $m,n$. From the asymptote equation $y=\pm bx/a$:

$Pm=Pn=\frac{bx}{a},Cm=Cn=\frac{x\sqrt{a^2+b^2}}{a}=FM+AC=GM-AC.$

Then the curve-to-asymptote intercepts are

$Mm=Nn=\frac{bx-ay}{a},Nm=Mn=\frac{bx+ay}{a},$

so

$\boxed{Mm\cdot Nm=Mm\cdot Mn=Nn\cdot Mn=Nn\cdot Nm=b^2.}$

The rectangle on the two intercepts of any chord between the curve and the two asymptotes is the constant $b^2=A{D}^2$ (source: chapter6, §160). This is the principal asymptotic property of the hyperbola — Apollonius’s theorem in algebraic form.

Drawing $Mr$ from $M$ parallel to one asymptote $Cd$ and $Cr$ along the other, the segments $Mr$ and $Cr$ satisfy

$Mr\cdot Cr=\frac{a^2+b^2}{4}.$

So if one extends from $A$ the line $AE$ parallel to one asymptote and meeting the other at $E$, then $AE=CE=\frac{1}{2}\sqrt{a^2+b^2}$ and $Mr\cdot Cr=AE\cdot CE$. This is the principal property of the relationship between a hyperbola and its asymptotes (source: chapter6, §160).

Hyperbola in asymptote coordinates: $xy=h^2$ (§161)

Take one asymptote as the axis of abscissas, the centre $C$ as origin, and ordinates parallel to the other asymptote. Let $CP=x$, $PM=y$. Then from the previous section $yx=(a^2+b^2)/4=h^2$, where $h=AE=CE$. So

$\boxed{xy=h^2,y=\frac{h^2}{x}.}$

The hyperbola appears as an inverse-square hyperbola in this oblique frame. As $x\to 0$, $y\to \infty$; as $x\to \infty$, $y\to 0$ — the two asymptotes are the coordinate axes themselves (source: chapter6, §161).

This is the simplest possible equation for the hyperbola, paralleling $y^2=2cx$ for the parabola.

Tangent bisection at point of contact (§§161-162)

In the asymptote-coordinate frame, draw a line $QMNR$ parallel to an arbitrary line $GH$ (with $GH$ also parallel to a chord direction), meeting the curve at $M$ and $N$ and the two coordinate axes at $Q$ and $R$. Let $CQ=t$ and $QM=u$. Setting up similar triangles and substituting into $xy=h^2$ yields the quadratic

$u^2-\frac{GH}{CH}tu+\frac{G{H}^2}{CH\cdot CG}{h}^2=0,$

whose two roots $QM,QN$ have sum $\frac{GH}{CH}t=QR$ and product $\frac{G{H}^2}{CH\cdot CG}{h}^2$.

Sum of roots equals $QR$ means $QM+QN=QR$, hence (source: chapter6, §162):

$QM=RN\text{and}QN=RM.$

The chord and the segment between asymptotes have the same midpoint. When $M$ and $N$ coincide — i.e. the line $QR$ is tangent to the curve — the tangent is bisected at the point of contact $Z$:

If $XY$ is tangent to the hyperbola at $Z$, then $Z$ is the midpoint of $XY$.

This gives an immediate construction of the tangent at any point $Z$: draw $ZV$ parallel to one asymptote, take $VY=CV$, then the line through $Z$ and $Y$ is tangent at $Z$.

The tangent intercepts on the two asymptotes satisfy $CX\cdot CY=a^2+b^2=C{D}^2=CD\cdot Cd$.

Constant rectangle property in asymptote coordinates (§163)

Since the rectangle $QM\cdot QN=(G{H}^2/CH\cdot CG)h^2$ depends only on the direction of $QR$ (through the constant $G{H}^2/CH\cdot CG$), not on the position of $QR$, the rectangle $QM\cdot QN$ on the curve-intercepts of any chord parallel to a fixed line is constant. We also have $QM\cdot QN=QM\cdot MR=QN\cdot NR$. For the tangent parallel to that direction (where $M=N$, half the segment between asymptotes is $q$):

$QM\cdot QN=QM\cdot MR=RN\cdot RM=RN\cdot NQ=q^2.$

This is a significant property of hyperbolas drawn within their asymptotes (source: chapter6, §163).

Lines crossing both branches (§§164-165)

The hyperbola has two diametrically opposed parts $IAi$ and $KBk$, so a line $Mqrn$ may meet not just one branch in two points but rather both branches. With $Cq=t,qM=u$, by the same similar-triangle argument:

$u^2+\frac{Gh}{Ch}tu-\frac{G{h}^2}{CG\cdot Ch}{h}^2=0.$

The two roots $qM$ and $-qn$ (the second negative because directed toward the other asymptote) have sum $-Gh/Ch\cdot t=-qr$ and product $-G{h}^2/(CG\cdot Ch)\cdot h^2$. From the sum, $qM=rn$ and $qn=rM$ — the chord between the two branches and the segment between the asymptotes still have the same midpoint. From the product:

$qM\cdot qn=qM\cdot rM=rn\cdot qn=rn\cdot rM=\frac{G{h}^2}{CG\cdot Ch}{h}^2.$

This rectangle is constant for any line $Mn$ parallel to $Gh$, regardless of whether the chord $Mn$ stays on one branch or crosses to the other (source: chapter6, §165).

Notable points

• Asymptote = tangent at infinity. Euler’s derivation in §158 makes the geometric content of “asymptote” precise: it is the limit of tangent lines as the point of contact recedes to infinity. This is consistent with the modern projective picture in which the hyperbola, tangent at infinity, and asymptote line all coincide on the line at infinity.
• Coordinate choice eliminates two parameters. In the principal-axes form the hyperbola needs $a$ and $b$. In the asymptote frame it needs only $h^2=(a^2+b^2)/4$ — and even that scales out under a hyperbolic rotation $x\mapsto kx,y\mapsto y/k$. The asymptote frame is the most economical possible coordinatisation of a hyperbola.
• Apollonius’s theorem in two lines. Apollonius’s classical proof of $Mm\cdot Mn=$ const requires careful case analysis; Euler dispatches it from the equation $y=\pm bx/a$ for the asymptote and the equation of the curve in three lines (§160). This is a paradigmatic example of why Euler thinks analytic geometry has superseded the synthetic tradition.
• Tangent bisection is an asymptote-coordinate phenomenon. The fact that the tangent to a hyperbola is bisected by its asymptote intercepts at the point of contact (§162) follows from $QM+QN=QR$ — the sum of roots of the quadratic in $u$ being independent of the chord position. This is a pure consequence of the form $xy=h^2$ being symmetric under $u\mapsto QR-u$, and it gives the simplest tangent construction known for the hyperbola.
• Same constant across both branches. §165 shows that the rectangle property $qM\cdot qn=$ const holds whether the chord stays on one branch or crosses to the other. This unifies the two branches into a single curve from the point of view of asymptote geometry — no special case for “transverse” vs. “longitudinal” chords. The product changes sign with branch-crossing only because of an oriented-segment convention; the unsigned rectangle is constant.
• Equilateral hyperbola pairs with the circle. The equilateral hyperbola ($a=b$, perpendicular asymptotes) is the analogue of the circle ($a=b$, the only ellipse with all radii equal). In modern terms both are the “round” representatives of their respective genera, and both have particularly simple analytic representations: $y^2+x^2=a^2$ and $xy=h^2$.

Figures

Figures 33–34

Related pages

• chapter-6-on-the-subdivision-of-second-order-lines-into-genera
• hyperbola
• classification-of-conics
• ellipse
• parabola
• diameter-of-conic
• chord-rectangle-property
• oblique-coordinates