Hyperbola

Summary: With the centre as origin and the principal axis as $x$-axis, the hyperbola has equation $y^2=\frac{b^2}{a^2}(x^2-a^2)$ — exactly the ellipse equation with $b^2$ replaced by $-b^2$. The curve has four branches going to infinity and no real conjugate axis; the foci $F,G$ on the principal axis at distance $\sqrt{a^2+b^2}$ from the centre satisfy $GM-FM=2a$ for every point $M$. The tangent at $M$ bisects the angle between the focal radii. Every other ellipse property transfers by the same $b^2\mapsto -b^2$ substitution.

Sources: chapter6 §§153-157; figure 33 (in figures33-34)

Last updated: 2026-04-26

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Set-up: centre as origin (§153)

A hyperbola has $\gamma >0$ in $y^2=\alpha +\beta x+\gamma x^2$. Translating to the centre kills the linear term, leaving $y^2=\alpha +\gamma x^2$. Without loss of generality (one can swap $x$ and $y$ to switch sign), take $\alpha <0$, write $\alpha =-\gamma a^2$ where $a=\sqrt{-\alpha /\gamma }$, and obtain

$y^2=\gamma x^2-\gamma a^2=\gamma (x^2-a^2).$

The ordinate is imaginary for $|x|<a$, so the curve has no points on the axis between $A$ and $B=(\pm a,0)$. For $|x|>a$ the ordinate grows continuously and reaches infinity, giving the four branches $AI,Ai,BK,Bk$ — the principal property of the hyperbola (source: chapter6, §153).

No real conjugate axis; imaginary semiaxis $bi$ (§154)

Setting $x=0$ in $y^2=\gamma (x^2-a^2)$ gives $y^2=-\gamma a^2<0$, so the centre $C$ does not lie on the curve and there is no real conjugate axis. To preserve formal similarity with the ellipse equation, take the conjugate semiaxis to be $bi$ (purely imaginary): then $\gamma a^2=b^2$, so $\gamma =b^2/a^2$, and

$\boxed{y^2=\frac{b^2}{a^2}(x^2-a^2).}$

This is the ellipse equation with $b^2\mapsto -b^2$. Every algebraic manipulation valid for the ellipse remains valid for the hyperbola under this substitution. Euler exploits the affinity throughout.

Foci and the difference-of-distances property (§§154-155)

In the ellipse the focus-centre distance is $\sqrt{a^2-b^2}$. With $b^2\to -b^2$ this becomes $\sqrt{a^2+b^2}$:

$CF=CG=\sqrt{a^2+b^2}.$

For $M=(x,y)$ on the curve, the focal radii are

$FM=\frac{x\sqrt{a^2+b^2}}{a}-a,GM=\frac{x\sqrt{a^2+b^2}}{a}+a,$

so

$\boxed{GM-FM=2a.}$

The difference of focal distances is constant and equal to the principal axis $AB=2a$ — the hyperbolic counterpart of the ellipse’s $FM+GM=2a$ (source: chapter6, §154).

Tangent and the bisected focal angle (§155)

The tangent at $M$ meets the principal axis at $T$ with $CT=a^2/x$ (the conic tangent ratio is independent of the species). Using $FM\cdot GM=(a^2{x}^2+b^2{x}^2-a^4)/a^2$, one gets

$MT=\frac{ay}{bx}\sqrt{FM\cdot GM},FT=\sqrt{a^2+b^2}-\frac{a^2}{x},GT=\sqrt{a^2+b^2}+\frac{a^2}{x}.$

The crucial proportion is $FT/FM=a/x$ and $GT/GM=a/x$, so $FT/FM=GT/GM$ and

$\angle FMT=\angle GMT.$

The tangent at $M$ bisects the angle $\angle FMG$ between the focal radii (source: chapter6, §155). (For the ellipse the corresponding tangent bisects the external angle, since $FM+GM$ is constant; for the hyperbola, since $GM-FM$ is constant, the tangent bisects the internal angle. The reflection law in either case sends a focal ray off the curve through the other focus, with the appropriate inward/outward sign.)

The extended diameter $CM$ is the oblique diameter bisecting every chord parallel to the tangent at $M$.

Auxiliary tangent properties (§§156-157)

Drop the perpendiculars $CQ$ from the centre and $FS,Gs$ from the foci to the tangent at $M$. Then by direct calculation:

• $TS\cdot Ts=CT\cdot PT$ — the rectangle on the focal-tangent feet equals the rectangle on $CT,PT$.
• $FS\cdot Gs=b^2$ — the rectangle on the perpendiculars from the two foci to the tangent equals $b^2$ (the same identity as in the ellipse!).
• $QS=Qs$ and $CS=a$ — the segment from centre to $S$ has length $CA$.
• If $FX$ is drawn from the focus parallel to the tangent meeting $CQ$ at $X$, then $CX=\sqrt{b^2+C{Q}^2}$.

Erecting perpendiculars at the vertices $A,B$ to meet the tangent at $V,v$:

$AV=\frac{b^2(x-a)}{ay},Bv=\frac{b^2(x+a)}{ay},AV\cdot Bv=b^2.$

The product of the vertex-perpendicular intercepts equals $b^2=FS\cdot Gs$ — the four perpendiculars (two from foci, two from vertices) have the same constant rectangle, an invariant generalising the ellipse identity (source: chapter6, §157).

Notable points

• The substitution $b^2\to -b^2$ is a unifying device. Every ellipse property has a hyperbola counterpart obtained by this single sign change. Distances become differences ($FM+GM=2a$ becomes $GM-FM=2a$); $\sqrt{a^2-b^2}$ becomes $\sqrt{a^2+b^2}$; the bounded curve becomes the unbounded one. This is the analytic statement of the well-known projective fact that ellipse and hyperbola are the same curve over the complex projective plane, distinguished only by the reality of intersections with the line at infinity.
• No conjugate axis on the curve. Since the centre lies between the two branches and the conjugate “axis” never meets the hyperbola, $b$ is not a length on the curve. Euler retains the symbol because $b^2=a^2\gamma$ controls every other quantity (semilatus rectum $b^2/a$, focal distance $\sqrt{a^2+b^2}$, conjugate semidiameters), and because the substitution $b^2\to -b^2$ from the ellipse demands it. The geometric meaning of $b$ emerges only in the asymptote analysis of §158, where $AD=b$ and the asymptote slope is $b/a$.
• Internal vs. external angle bisection. The tangent to an ellipse bisects the external angle of the focal radii (so a billiard ball at $F$ bounces off the wall to $G$); the tangent to a hyperbola bisects the internal angle (so a ray toward $F$ reflects as if continuing from $G$). Both are consequences of differentiating $FM\pm GM=2a$ along the curve.
• $FS\cdot Gs=b^2$ is universal among conics with foci. The same identity holds for the ellipse. Together with $CS=a$, this is the projective core of the focal apparatus, surviving the genus split.
• No infinite-axis limit yet. Unlike the parabola (which Euler obtains as the $a\to \infty$ limit of the ellipse), there is no comparable degenerate limit of the hyperbola in §§153–157. The unique structural feature of the hyperbola — the asymptotic behavior — is treated separately in §§158-165.

Figures

Figures 33–34

Related pages

• chapter-6-on-the-subdivision-of-second-order-lines-into-genera
• classification-of-conics
• ellipse
• parabola
• asymptotes-of-hyperbola
• principal-axes-and-foci
• conjugate-diameters
• tangent-properties-conic