My point is that the ellipses for c > 0 do not approach the line segment [-1,1]. They approach the line segment [-sqrt(⅓), sqrt(⅓)]. --Dan From: Henry Baker <hbaker1@pipeline.com> Subject: Re: [math-fun] Q re ellipses in the complex plane Date: 18 February, 2013 5:19:14 PM PST ----- . . . I hadn't thought it possible that an ellipse could approach a line segment while still keeping its foci away from the line segment ends, but I was wrong! . . . ----- On 2013-02-19, at 4:33 PM, Henry Baker wrote:

Re: "This is not what happens":

Tell that to Maxima. P(z) has roots at +-1, +ci.

P'(z) has roots at +ci/3+-sqrt(3-c^2)/3 = +-1/sqrt(3) when c=0.

(%i1) declare(z,complex); (%o1) done (%i2) P(z):=(z-1)*(z+1)*(z-%i*c); (%o2) P(z) := (z - 1) (z + 1) (z - %i c) (%i3) solve(P(z),z); (%o3) [z = %i c, z = - 1, z = 1] (%i4) solve(diff(P(z),z),z); 2 2 sqrt(3 - c ) - %i c sqrt(3 - c ) + %i c (%o4) [z = - -------------------, z = -------------------] 3 3 (%i5) %,c=0; 1 1 (%o5) [z = - -------, z = -------] sqrt(3) sqrt(3) (%i6) %,numer; (%o6) [z = - 0.57735026918963, z = 0.57735026918963]

I.e., these numbers are not +-1.

At 03:33 PM 2/19/2013, Dan Asimov wrote:

...

This is not what happens.

The limit of the ellipse as c -> 0 is indeed a degenerate ellipse that is precisely the line segment between its foci, +-sqrt(â…“).