%% focus4.tex - display 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}
\thispagestyle{empty}

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

  \useasboundingbox ([scale=3/2] 5,4) to ([scale=-1] \tikztostart) ;

  \def\Q{current bounding box}
  \draw[->] (\Q.west) -- (\Q.east) node[right] {$x$} ;
  \draw[->] (\Q.south) -- (\Q.north) 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,radius=3pt] (3,0) circle[] (-3,0) circle[] ;
  \fill[red,radius=3pt] (0,3) circle[] (0,-3) circle[] ;

  \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) {$-ic$} ;

\end{tikzpicture}

\end{document}

%%%%%%%%

%% png creation (normal size, transparent)

%%% pdflatex focus4.tex
%%% pdftocairo -png -singlefile -transp -r 180.675 focus4.pdf tmp
%%% magick -define png:exclude-chunk=date,time tmp.png -crop 615x509+307+250 +repage focus4.png
%%% optipng focus4.png

%%%% Resolution 180.675 (=72.27/0.4) ppi translates 0.4pt (line width)
%%%% to 1 pixel.  \parindent=12pt translates to 30 pixels.

%%%% magick adds white (fallback) background to the transparent image.

%%%% trim parameters:
%%% magick tmp.png -format "%@\n" info: %% 557x458+347+261

%%%% Padding: get origin in the center + padding 10px.

%%%% Image size is 615x509.

%% 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)

%%% pdftocairo version 0.86.1
%%% ImageMagick 7.1.2-19 Q16-HDRI x86_64 23897 https://imagemagick.org
%%% OptiPNG version 0.7.7

%%%%%%%%

%% Created:
%% 2026-09-02
%% Latest change:
%% 2026-09-07

%%%%%%%%