%% focus4.tex - display of the focus hyperbola
%% Author: Markus Rost <https://www.math.uni-bielefeld.de/~rost/>
%% Created: Sep 2 2026

%% to be compiled with pdflatex

\documentclass{amsart}

\usepackage{tikz}

\begin{document}

\begin{displaymath}
  \begin{tikzpicture} [scale=1/2]

    \draw[->] (-5*3/2,0) -- (5*3/2,0) node[right] {\rlap{$x$}} ;
    \draw[->] (0,-4*3/2) -- (0,4*3/2) node[above] {%
      $y\color{red}\rlap{$\;iy$}$} ;

    \draw[red,domain=-40:40]
      plot ({5*sec(\x)},{4*tan(\x)})
      plot ({-5*sec(\x)},{4*tan(\x)}) ;
    \draw (0,0) circle[x radius=5,y radius=4] ;

    \draw[magenta,semithick] (0,0) circle[radius=3] ;
    \draw[blue, thick, domain=-60:60]
      plot ({3*sec(\x)},{3*tan(\x)})
      plot ({-3*sec(\x)},{3*tan(\x)}) ;

    \fill[blue] (3,0) circle[radius=3pt] ;
    \fill[blue] (-3,0) circle[radius=3pt] ;
    \fill[red] (0,3) circle[radius=3pt] ;
    \fill[red] (0,-3) circle[radius=3pt] ;

    \node[above right,yshift=-1pt)] at (3,0) {$c$} ;
    \node[above left,yshift=-1pt] at (-3,0) {$\smash{-}c$} ;
    \node[above left,yshift=-1pt)] at (0,3) {$ic$} ;
    \node[below left,yshift=2pt)] at (0,-3) {$\smash{-}ic$} ;

  \end{tikzpicture}
\end{displaymath}

An ellipse together with its axes, its focus hyperbola and the $4$
focus points ($2$ real, $2$ imaginary).  Drawn in the $(x,y)$-plane
and in the $(x,iy)$-plane.  In the latter, the ellipse equation yields
an hyperbola and the focus hyperbola becomes a circle.

\begin{gather*}
  \frac{x^2}{25}+\frac{y^2}{16}=1 ,\qquad xy=0 ,\qquad
  \frac{x^2-y^2}{9}=1
  \\
  (a,b,c)=(5,4,3)
\end{gather*}

\end{document}

%% program versions

%%% This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Debian) (preloaded format=pdflatex)
%%% Document Class: amsart 2017/10/31 v2.20.4
%%% Package: tikz 2020/01/08 v3.1.5b (3.1.5b)

%% Created:
%% 2026-09-02