Tangent on ellipse in a point with tikz

Question

In this MWE there is tangent with help of the answer of @Jake in How do I make tangents to ellipses and lines parallel to these? But I do not succeed in using this point that I got by tangent=0.35 later on in order to connect it with F1 and F2. In other words: The blue and the black point on the ellipsis line should be the same but how can I get it exact without guessing the angle of the point where I've drawn the tangent?

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{calc,decorations.markings,intersections}
\usepackage{pgfplots}
\pgfplotsset{compat=1.15}
\begin{document}
\begin{tikzpicture}[
  dot/.style={draw,fill,circle,inner sep=1pt},
  tangent/.style={
        decoration={
            markings,% switch on markings
            mark=
                at position #1
                with
                {
                    \coordinate (tangent point-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (0pt,0pt);
                    \coordinate (tangent unit vector-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (1,0pt);
                    \coordinate (tangent orthogonal unit vector-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (0pt,1);
                }
        },
        postaction=decorate
    },
    use tangent/.style={
        shift=(tangent point-#1),
        x=(tangent unit vector-#1),
        y=(tangent orthogonal unit vector-#1)
    },
    use tangent/.default=1
  ]
  \def\a{4} % major half axis
  \def\b{2} % minor half axis
  \draw[thick, tangent=0.35] (0,0) ellipse ({\a} and {\b});
  \fill (tangent point-1) circle [radius=2pt];
  \draw [use tangent] (2,0) -- (-2,0);
  \def\angle{125} % angle for point on ellipse
  %Labelling the foci
  \node[dot,label={below:$F_1$}] (F1) at ({-sqrt(\a*\a-\b*\b)},0) {};
  \node[dot,label={below:$F_2$}] (F2) at ({+sqrt(\a*\a-\b*\b)},0) {};
  %Point on ellipsis
  \node[dot,label={\angle:$P$},blue] (P) at (\angle:{\a} and {\b}) {};
  \draw (F1) -- (P) (P) -- (F2);
\end{tikzpicture}
\end{document}

Answer 1

With a bit of help from tkz-euclide and in particular the bisector out macro, this is pretty straightforward. This is achievable in simple TikZ with calc library but here is a simple line to obtain the tangent.

\documentclass[tikz,border=3.14mm]{standalone}

\usepackage{tkz-euclide}
\usetikzlibrary{calc}

\begin{document}
    \begin{tikzpicture}[dot/.style={draw,fill,circle,inner sep=1pt},]
        \def\a{4} % major half axis
        \def\b{2} % minor half axis
        \draw[thick] (0,0) ellipse ({\a} and {\b});
        
        \node[dot,label={below:$F_1$}] (F1) at ({-sqrt(\a*\a-\b*\b)},0) {};
        \node[dot,label={below:$F_2$}] (F2) at ({+sqrt(\a*\a-\b*\b)},0) {};
        \def\angle{125} % angle for point on ellipse
        \node[dot,label={\angle:$P$},blue] (P) at (\angle:{\a} and {\b}) {};
        \draw (F1) -- (P) -- (F2);
        
        \tkzDefLine[bisector out](F1,P,F2) \tkzGetPoint{K}
        \draw ($(K)!-2!(P)$)--($(P)!-2.5!(K)$);
          
    \end{tikzpicture}
\end{document}

I usually prefer to code everything in tkz-euclide when possible, but I thought that you would keep your original code instead.

Answer 2

While you are waiting for the TikZ cavalry, here is one I did earlier in Metapost for comparison.

Metapost has the useful concept of (fractional) points along a path. In the case of an ellipse there are 8 points starting at "3 o'clock".

• point t of E gives you a pair of coordinates for the point
• direction t of E gives you a pair that represents the tangent at that point

To get a nice line to draw I have used angle to find the rotation of the direction and applied it to a known line...

\documentclass[border=5mm]{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
vardef focus(expr e, n) = 
  numeric a,b;
  a = arclength (point 0 of e .. center e);
  b = arclength (point 2 of e .. center e);
  ((a+-+b)/a)[center e, point if n=1: 4 else: 0 fi of e]
  enddef;

beginfig(1);

path E; E = fullcircle xscaled 233 yscaled 144 rotated 8; 
numeric t; t = 2.7182818;

z0 = point t of E;
z1 = focus(E,1);
z2 = focus(E,2);

path tangent; 
tangent = (left--right) scaled 42 
        rotated angle direction t of E
        shifted point t of E;

drawoptions(withcolor 3/4);
draw point 0 of E -- point 4 of E withcolor .7 white;
draw point 2 of E -- point 6 of E withcolor .7 white;

drawoptions(withcolor 3/4 red);
draw tangent;
draw z1 -- z0 -- z2;

drawoptions(withcolor 5/8 blue);
draw E;

drawoptions();
dotlabel.lrt(btex $F_1$ etex, z1);
dotlabel.lrt(btex $F_2$ etex, z2);
dotlabel.ulft(btex $P$   etex, z0);

endfig;
\end{mplibcode}
\end{document}

The supplied focus routine does the arithmetic to find you the focus points given an elliptical path. This is wrapped in luamplib so compile with lualatex.

Answer 3

A geometric method use the fact that the normal line at a point on a ellipse is the internal bisector of the angle from that point to two foci. A TikZ way is in here with the user-defined command bisectorpoint.

In Asymptote, bisector direction is a built-in command.

pair dir(path p, path q)
returns unit(dir(p)+dir(q)).

In particular,

pair Mn=M+dir(M--F1,M--F2); 
pair Mt=rotate(90,M)*Mn;

give the point Mn on the normal line and the point Mt on the tangent line at M of the ellipse.

unitsize(1cm);
real a=3, b=2;
real c=sqrt(a^2-b^2);
pair F1=(-c,0), F2=(c,0);
real t=130;
pair M=(a*Cos(t),b*Sin(t));  // a point on the ellipse
// M--Mn is the internal bisector of MF1 and MF2 
pair Mn=M+dir(M--F1,M--F2); 
pair Mt=rotate(90,M)*Mn;
dot(F1^^F2,gray);

draw((a+1,0)--(-a-1,0)^^(0,b+.5)--(0,-b-.5),gray);
draw(F1--M--F2,gray);
draw(ellipse((0,0),a,b));
draw(interp(M,Mt,-2)--interp(M,Mt,1.5),red);
draw(interp(M,Mn,-1)--interp(M,Mn,1.5),blue);

label("$F_1$",F1,S);
label("$F_2$",F2,S);
label("$M$",M+.4*dir(165));

Answer 4

Using tzplot:

\documentclass[border=1mm]{standalone} 

\usepackage{tzplot}

\begin{document}

\begin{tikzpicture}[scale=1.4]
\def\a{4} % major half axis
\def\b{2} % minor half axis
\tzbbox(-4,-2)(4,2.2)
\settzdotsize{4pt}
\tzellipse[thick]"AA"(0,0)({\a} and {\b})
\tzvXpointat*{AA}{-2.5}(P){$P$}
\tztangent{AA}(P)[-4:-1]
\tzcoors*({-sqrt(\a*\a-\b*\b)},0)(F1){$F_1$}[b]({+sqrt(\a*\a-\b*\b)},0)(F2){$F_2$}[b];
\tzlines(F1)(P)(F2);   
\end{tikzpicture}    

\end{document}