Ellipse

Summary: With the centre as origin and the principal axis as $x$-axis, the ellipse has equation $y^2=\frac{b^2}{a^2}(a^2-x^2)$. The two foci on the major axis at distance $\sqrt{a^2-b^2}$ from the centre satisfy $FM+GM=2a$ for every point $M$ on the curve; the tangent at $M$ makes equal angles with the focal radii $FM$ and $GM$. Conjugate semidiameters $p,q$ at angle $s$ obey $p^2+q^2=a^2+b^2$ and $pq\sin s=ab$. Moving the origin to the vertex gives $y^2=2cx-\frac{c(2d-c)}{d^2}{x}^2$ with $c$ the semilatus rectum and $d$ the vertex-focus distance.

Sources: chapter6 §§138-147; figure 31 (in figures30-32)

Last updated: 2026-04-26

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Set-up: centre as origin (§138)

Starting from $y^2=\alpha +\beta x-\gamma x^2$ (with $\gamma >0$ since the sign convention for an ellipse makes the leading coefficient negative), translate the abscissa by $\beta /(2\gamma )$ so the centre $C$ becomes the origin. The equation becomes

$y^2=\alpha -\gamma x^2.$

Setting $y=0$ gives $x=\pm \sqrt{\alpha /\gamma }$ — the curve crosses the axis at two points $A,B$ with $CA=CB=\sqrt{\alpha /\gamma }$. Setting $x=0$ gives $y=\pm \sqrt{\alpha }$ — two more points $D,E$ with $CD=CE=\sqrt{\alpha }$. Outside $|x|>\sqrt{\alpha /\gamma }$ the ordinate is imaginary, so the curve is bounded.

Write the principal semiaxis $CA=CB=a$ and the conjugate semiaxis $CD=CE=b$. Then $\alpha =b^2$ and $\gamma =b^2/a^2$, giving the canonical form (source: chapter6, §138):

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

When $a=b$ the ellipse degenerates to a circle $y^2+x^2=a^2$; otherwise it is oblong. By convention the major axis is taken as the $x$-axis: $a>b$.

Foci and the sum-of-distances property (§§139-140)

The foci $F$ and $G$ lie on the major axis at distance $CF=CG=\sqrt{a^2-b^2}$ from the centre. The semilatus rectum (semiparameter) is the ordinate at either focus:

$\frac{b^2}{a}.$

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

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

so

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

The sum of the focal radii at any point of the ellipse equals the major axis. This is the famous string-construction property — pin string of length $2a$ to the two foci, trace with a pencil keeping the string taut, and one obtains the ellipse exactly (source: chapter6, §140).

Tangent and the equal-focal-angle property (§§141-142)

The tangent $TMt$ at $M$ meets the major axis at $T$ and the minor axis at $t$. From the conic tangent ratio $CP/CA=CA/CT$ one gets $CT=a^2/x$, and analogously $Ct=b^2/y$. Then

$TP=\frac{a^2-x^2}{x}=\frac{a^2{y}^2}{b^{2}x},TM=\frac{y\sqrt{b^4{x}^2+a^4{y}^2}}{b^{2}x},$

with the corresponding sine, cosine, and tangent of $\angle CTM$ following directly. From $FT/FM=a/x$ and $GT/GM=a/x$ Euler obtains the central focal property (source: chapter6, §142):

$\sin \angle FMT=\sin \angle GMt,$

so the tangent at $M$ makes equal angles with the two focal radii $FM$ and $GM$. (Reflection law: a ray from one focus reflects off the curve toward the other.)

Auxiliary tangent constructions (§§143-144)

• $CS\parallel GM$ meeting the tangent at $S$ gives $CS=a$ — the segment of any line through the centre parallel to a focal radius and stopped at the tangent equals the principal semiaxis.
• The perpendicular $FS$ from a focus to the tangent and the perpendicular $Gs$ from the other focus satisfy $FS\cdot Gs=b^2$ — the rectangle on the focal perpendiculars to the tangent equals $b^2$, the square of the minor semiaxis.
• $CQ$ perpendicular to the tangent satisfies $CQ\cdot \sqrt{(FM\cdot GM)}=ab$.

Conjugate diameter laws (§§145-146)

For a point $M$ on the curve, let $CM=p$ be the semidiameter through $M$ and $CK=q$ the conjugate semidiameter (parallel to the tangent at $M$), with included angle $s=\angle MCK$. Then (source: chapter6, §145):

$p^2+q^2=a^2+b^2,pq\sin s=ab.$

The first identity is the constant sum of squares of conjugate semidiameters; the second says that the parallelogram inscribed around a conjugate pair has constant area $4ab$. The orthogonal pair (the principal axes themselves) is the unique pair with $s=90°$, and that pair maximises $|a-b|$ — i.e. the orthogonal conjugates differ from each other the most (source: chapter6, §146).

The equal-conjugate pair is found by setting $p=q$: then $p=q=\sqrt{(a^2+b^2)/2}$, $\sin s=2ab/(a^2+b^2)$. The semi-angles satisfy $\tan (s/2)=a/b$, and the equal conjugate semidiameters $CM,CK$ are parallel to the chords $AE$ and $BE$ joining vertex to minor-axis endpoint.

Vertex-origin form and the latus rectum (§147)

Take instead the vertex $A$ as origin: substitute $a-x$ for $x$. The equation becomes

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

The coefficient $2b^2/a$ is called the parameter or latus rectum of the ellipse. Writing $c=b^2/a$ for the semilatus rectum and $d=AF=a-\sqrt{a^2-b^2}$ for the vertex-focus distance, one gets $a=d^2/(2d-c)$ and the vertex form (source: chapter6, §147):

$\boxed{y^2=2cx-\frac{c(2d-c)}{d^2}{x}^2.}$

For real ellipses $2d>c$ always, since $a=d^2/(2d-c)$ must be positive.

The vertex form is the version Euler will degenerate to a parabola in the next section by setting $2d=c$.

Notable points

• Algebraic, not metric, foundation. The foci are not introduced as the points whose distance-sum is constant; they are derived (§139) by computing $\sqrt{a^2-b^2}$ from the canonical equation. The distance-sum property (§140) is then a consequence. This inverts the modern “metric definition first” pedagogy and matches Euler’s earlier definition of foci in chapter 5 §128 as the points from which $DM$ is rational in $x$.
• The two conjugate-diameter identities are the heart of ellipse geometry. $p^2+q^2=a^2+b^2$ and $pq\sin s=ab$ together let any conjugate-pair calculation be reduced to the principal axes. They generalise Pythagoras (which is the case $s=90°$, $p=a$, $q=b$) and the rectangle-on-axes ($ab$ is the area of the principal rectangle, $pq\sin s$ that of the conjugate-aligned parallelogram).
• Equal focal perpendiculars give $b^2$. $FS\cdot Gs=b^2$ is the projective analogue of $FM\cdot GM\ne$ const: although the focal radii vary, the perpendiculars from the foci to the tangent multiply to a constant. This is the foundation of the Newtonian focal-chord constructions.
• Vertex form bridges to the parabola. Writing the equation about the vertex with parameter $c$ and vertex-focus distance $d$ pulls $a$ off-stage. The substitution $2d=c$ — equivalent to $a=\infty$ — collapses the equation to $y^2=2cx$, the parabola. So the parabola is literally an ellipse with infinite major axis (source: chapter6, §148).

Figures

Figures 30–32

Related pages

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