On the constructibility with ruler and compass of a minimum chord in a parabola Page: 11
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When there is only one distinct real root, it repr tas with certainty the slope of the
minimum chord; when the are two distinct real roots, the simplemon correponds to the
minimum chord, and when there are three distianct real oots the slope of the minimum
chrd is either the smalt or the largest of the oots, or both. At any rate, knowing the
number of real roots does not help in wairthti- , w marked ruler and compassm,
the m.nm chlrd. Iadeed, aeacrding to the f.adlmtal the of r constctibiliy
with uamard ruler and omp ass, IS l 5], it one dmoa by & the subfield of R
generated by ,s, y, the degree [(() : k) of the ield en ion k(0) of , obtained by
adoini to ka rea mlroot 0 of Equation 1, must beo a poerof 2, in order for (and
thus the minimum chord) to be constrctible. Or, it is easy to find (rational) numbers
c, s, p such that this degree is 3, e.g., c = s = p = 1. We want to stress here that we
look for a geometric way to construct the minm.m cord mc Otherwie americal
methods can be sd for any given (e,v) to appridmate the solutions of Equation 1,
and determine which one correspond to a minim chrd by evuting the function f
at them.
Let as now $xa slope a 0. From Equation 1, all points (s,p ) which admit a
mi.nim- chord of lopes ar on the line
(2) -(3 + )X + 2sY + 2cs(2' +. 1) = 0.
More preisely, they belong to the straight segment whose endpoints are obtained by in-
tersetin the lin give by equation 2 with the parabola P It > 0 their X-oordinates
a 2cs and Sc(2@ + 1)/s. If = 0 theybelong to the -a part of the Y-axis.
Altertiely, a csmary condition for a minimum chord through (s, ) to have slope a,
S>O, is
33 + 1 2. + 1
(3) y= s s- - (2 + 1), 2c s < 2c---.
Not all the points of this segment admit minimum chords with slopes, but only a
sutmment of it. The follow proposition characterize this subsegm t.
ProposItion 4. A point (8, 1) inside the parol P admit. a minimum chord with slope
a if and only if
a) s=, for.=0.
b) p a - c(2 + 1), 2,csz S 2c Lc , for s > 0.
Proof. Clearly a) is true, since if = 0 Equation I has m = 0 as its only real solution.
b) If (z, y) satiss Equation 3 for some s > 0, then
f(m)- f(S) = ( -m -s)('e' + -(2cs - s)m + c - .
Evidently f(s) is a minimum value for f(m) if and only if the discriminant ofthe quadratic
polynomia in m,
em' + (2cs - )m + s ' - 7 +
2 2.
is not poittve, Le., sz( - 2 - 2c/s) O0. This inequality, cmbia with the resriction
on a given by Eqution 3, proves b). O
This analytic finding translate neatly into constructive geometryNicolae Angsel
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Anghel, Nicolae. On the constructibility with ruler and compass of a minimum chord in a parabola, article, 1997; [Berkeley, California]. (https://digital.library.unt.edu/ark:/67531/metadc177421/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.