Two-Ordinate Sum and Product

Summary: The starting point of chapter-16-on-finding-curves-from-properties-of-the-ordinate (§§364–366). For any curve $y^2-Py+Q=0$ with $P,Q$ non-irrational in $x$, the two ordinates for each abscissa have sum $P$ and product $Q$ (Vieta). Constancy of the sum gives a diameter, constancy of the product gives the chord-product property, and §365 packages all four conic species under one family. §366 records Pierre Variot’s prize-winning property of central conics.

Sources: chapter16 (§§364–366). Figure 19 (in figures19-22, hyperbola with asymptote as axis), figure 77 (in figures76-80, central conic with $\pi /4$ chords).

Last updated: 2026-05-11.

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The Vieta setup (§364)

For an algebraic curve of the form

$y^2-P(x)y+Q(x)=0$

where $P,Q$ are non-irrational functions of $x$, each abscissa $x$ has either two real ordinates $y_1,y_2$ or none. By Vieta’s relations,

$y_1+y_2=P(x),y_1\cdot y_2=Q(x).$

Sum and product of the two ordinates are encoded directly in the coefficients. Any symmetric function of $y_1,y_2$ becomes a polynomial in $P,Q$ via Newton’s identities, which is what turns “inverse” problems (specify a symmetric property, recover the curve) into algebraic manipulation.

The four base cases

Coefficient condition	Implication on ordinates	Geometric meaning
$P=$ const	$y_1+y_2=$ const	orthogonal diameter perpendicular to the axis (diameter-of-conic)
$P=a+nx$	sum is linear in $x$	oblique diameter $z=a/2+nx/2$ where ordinates are bisected
$Q=$ const	$y_1\cdot y_2=$ const	curve never meets the axis (the limiting case where $y_1{y}_2\to 0$ would put a root on it)
$Q=\alpha +\beta x+\gamma x^2$ factors as $\alpha +\beta x+\gamma x^2=\gamma (x-x_1)(x-x_2)$	$y_1{y}_2/(x-x_1)(x-x_2)=\gamma$, constant	chord-product property: ratio of ordinate-product to interval-product is constant (chord-rectangle-property)

§365 highlights that the product-of-ordinates / product-of-intervals theorem of chord-rectangle-property — originally derived for conic sections — actually holds for any curve $y^2-Py+Q=0$ with $Q$ a quadratic in $x$ that factors. Taking the chord $EF$ that meets the curve at $E,F$ as the axis (figure 19, with the hyperbola asymptote example), the product $PM\cdot PN$ has a constant ratio to $PE\cdot PF$ across all parallel chords. Many higher-order curves inherit this property.

The unified conic family (§365)

Letting $P=b+nx$, the equation $y^2-Py+Q=0$ takes the form

$y^2-nxy+x^2=ax+by$

(after rescaling). This equation gives all four conic species at once:

$n$	Species	Geometry
$n=0$	circle	ordinates perpendicular to the axis, right-angle property at $EPM$
$0<n^2<4$	ellipse	bounded, two intersections with parallel-chord direction
$n^2=4$	parabola	borderline, asymptotic to a direction
$n^2>4$	hyperbola	two-branched, two asymptote directions

The case-split is exactly the discriminant $n^2-4$ of the quadratic in the highest-degree terms — same trichotomy as §131’s $\gamma$-sign in classification-of-conics, just in a different normalization.

Variot’s prize property (§366)

For any central conic $AEBF$ with principal axes $AB$ and $EF$ (figure 77), draw two straight lines $pq$ and $mn$ each inclined at $\pi /4$ (“half a right angle”) to the principal axis $AB$, meeting at an interior point $h$. Then

$mh\cdot nh=ph\cdot qh.$

In particular, taking the two lines $PQ$ and $MN$ through the center $C$ at $\pm \pi /4$: $MC\cdot NC=PC\cdot QC$. Since this holds for every interior point $h$, any pair of parallel translates of the two $\pi /4$ chords inherits the same equal-rectangle relation. Pierre Variot won an Académie prize for this theorem, hence the “prize-winning” tag (source: chapter16, §366).

A weaker variant of the same property: if $MN$ and $PQ$ are drawn at the same angle to $AB$ (so $PCA=NCA$, hence $CP=CN$), then again $mh\cdot hn=ph\cdot hq$. This case is just the chord-rectangle property of chord-rectangle-property applied to two equally-inclined chords; the genuine prize property is the $\pi /4$ specialization.

Reduction template for symmetric conditions

The pattern of §§364–366 is the chapter 16 template, repeated in higher strands:

1. Write the condition on $y_1,y_2$ as a symmetric polynomial.
2. Use Newton’s identities to convert it to a polynomial relation in $P,Q$.
3. The relation pins down one of $P,Q$ in terms of the other; the remaining one is freely chosen as a function of $x$, generating an infinite family.
4. The curve equation is then $y^2-P(x)y+Q(x)=0$ with the chosen $P$ (or $Q$).

For sum = const this fixes $P$, leaving $Q$ free. For product = const this fixes $Q$, leaving $P$ free. The strand-2 and later analyses extend this to higher symmetric polynomials.

Figures

Figures 19–22

Figures 76–80

Related pages

• chapter-16-on-finding-curves-from-properties-of-the-ordinate — chapter summary.
• diameter-of-conic — chapter 5 forward direction: sum of roots gives the diameter.
• chord-rectangle-property — chapter 5 forward direction: product of roots gives the chord-rectangle.
• classification-of-conics — chapter 6 discriminant-sign trichotomy that matches the §365 unified family.
• sum-of-ordinate-powers-curves — §§367–373 extension to higher power sums.