ordinate-powers-from-fixed-point

Ordinate Powers From a Fixed Point

Summary: §§419–425: the chapter-16 power-sum machinery, redirected to chord-distances. $C{M}^2+CM\cdot CN+C{N}^2=a^2$ is solved by a §419-style framework; $C{M}^n+C{N}^n=a^n$ has genuine solutions only for even $n$ (odd $n$ requires squaring, which doubles intersections). The closed-form solution via the §372 trick (in sum-of-ordinate-powers-curves) yields $z^n=a^n{L}^n/(L^n+(L-M{)}^n)$; §425 extends to mixed-symmetric-function generalizations.

Sources: chapter17 §§419–425.

Last updated: 2026-05-11.

---

$C{M}^2+CM\cdot CN+C{N}^2=a^2$ (§§419–420)

This is the $n=1$ case of the §412 family $C{M}^2+nCM\cdot CN+C{N}^2=a^2$. Substituting $P=Mz/L$, $Q=N{z}^2/L$:

$a^2{L}^2=(x^2+y^2)(L^2-LM+M^2)$

with $L$ of degree $m+1$, $M$ of degree $m$. A second general form, using $L=(x^2+y^2)N$, is

$a^2(x^2+y^2)N^2=(x^2+y^2{)}^2{N}^2-(x^2+y^2)MN+M^2,$

with $M$ one degree higher than $N$.

§420 lists the simplest cases:

• $M=0$: $a^2=x^2+y^2$ — circle centered at $C$.
• $M=b$, $L=x$ (so $m=1$, lowest non-trivial): $a^2{x}^2=(x^2+y^2)(x^2-bx+b^2)$, i.e.

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

• $N=1$, $M=bx$ from the second form: $a^2(x^2+y^2)=(x^2+y^2{)}^2-bx(x^2+y^2)+b^2{x}^2$, solvable to

$x^2+y^2=\frac{1}{2}bx+\frac{1}{2}{a}^2\pm \sqrt{\frac{1}{4}{a}^4+\frac{1}{2}{a}^{2}bx-\frac{3}{4}{b}^2{x}^2}.$

$C{M}^n+C{N}^n=a^n$ (§§421–425)

§421 sets up the general $C{M}^n+C{N}^n=a^n$ problem via Newton’s identity $\sum z_i^n$ in terms of $P,Q$. For $n=3$: $C{M}^3+C{N}^3=P^3-3PQ$, but $P^3$ and $PQ$ are both irrational (carry $z$ as $\sqrt{x^2+y^2}$), so demanding $P^3-3PQ=a^3$ has no two-intersection solution.

§422 with $n=4$ uses $P^4-4P^{2}Q+2Q^2=a^4$, set $Q=P^2+\sqrt{\frac{1}{2}{P}^4+\frac{1}{2}{a}^4}$, etc. — clean, no irrationality, because the square root sits under sign-controllable squaring.

The pattern Euler extracts: $C{M}^n+C{N}^n=a^n$ has solutions iff $n$ is even.

The §423 shortcut: §372 redux

§423 points back to §372 of sum-of-ordinate-powers-curves: rather than expanding $P^n+...$, just use $CM\cdot CN=Q$ to write $CN=Q/CM=Q/z$. Then $C{M}^n+C{N}^n=a^n$ becomes

$z^n+\frac{Q^n}{z^n}=a^n⟺z^n+\frac{N^n{z}^n}{L^n}=a^n⟺z^n=\frac{a^n{L}^n}{L^n+N^n}.$

Combining with $L-M+N=0$ (so $N=M-L$):

$\boxed{z^n=(x^2+y^2{)}^{n/2}=\frac{a^n{L}^n}{L^n+(L-M{)}^n}=\frac{a^n{L}^n}{L^n+(M-L{)}^n}.}$

For even $n$: rational, satisfies the desired condition.

For odd $n$: removing the irrationality $z^n=z\cdot z^{n-1}$ requires squaring; the resulting curve has four intersections per line through $C$, not two — strictly speaking no solution.

The §410 confirmation: $n=2$ gives $z^2=x^2+y^2=a^2{L}^2/(L^2+(L-M{)}^2)$, which matches the boxed equation of constant-chord-interval-conchoid up to substitution.

§424 — same answer from the sum-of-chords path

§424 derives the same closed form via $CM+CN=P$: with $CM=z$ and $CN=P-z$, we want $z^n+(P-z{)}^n=a^n$. Substituting $P=Mz/L$ and the same simplification:

$z^n+\frac{z^n(M-L{)}^n}{L^n}=a^n⟺z^n=\frac{a^n{L}^n}{L^n+(M-L{)}^n},$

confirming §423’s formula. The two derivations are dual; either symmetric function $P$ or $Q$ leads to the same parametrization.

§425 — mixed-symmetric-function generalization

The most general form §425 handles in one stroke is

$C{M}^n+C{N}^n+\alpha CM\cdot CN(C{M}^{n-2}+C{N}^{n-2})+\beta C{M}^{2}C{N}^2(C{M}^{n-4}+C{N}^{n-4})+\cdots =a^n.$

Setting $CM=z$, $CN=Q/z=Nz/L$:

$z^n(L^n+N^n)+\alpha LN(L^{n-2}{N}^2+N^{n-2}{L}^2)+\cdots =a^n{L}^n,$

which combined with $L-M+N=0$ (taking $L=M-N$ or $N=M-L$) yields infinitely many solutions for each choice. The take-home is that any symmetric polynomial in $CM,CN$ equated to a constant fits into the Vieta-plus-$L-M+N=0$ framework, so there’s no further difficulty.

Why “even-$n$ only”

The non-irrational structure: $P/z$ and $Q/z^2$ are non-irrational of degrees $-1,-2$. So $z^k{P}^{\ell }$ stays non-irrational iff the power of $z$ (after expanding) is even. Power sums $\sum z_i^n$ are polynomials in $P,Q$ of total degree $n$ in $z$ — i.e. $z^n$ overall — so $n$ even gives a non-irrational identity, $n$ odd gives an irrational one.

Related pages

• chapter-17-on-finding-curves-from-other-properties
• sum-and-product-of-chords-from-point — the $n=1$ degenerate case.
• constant-chord-interval-conchoid — the $n=2$ case viewed from a different angle.
• sum-of-ordinate-powers-curves — the §372 result this redirects.
• two-ordinate-sum-and-product — the chapter-16 Vieta setup.
• three-point-equation-from-fixed-point — extends the same machinery to three chord-distances.