How can I draw an ellipse so that four vertices don't lie on x -axes and y -axes?

Question

I read at here How to draw an ellipse?. If the equation of ellipse is x^2/25+y^2/144=1, I tried

\documentclass[border=2]{standalone}
\usepackage{tikz}
\begin{document}
\newcommand{\boundellipse}[3]% center, xdim, ydim
{(#1) ellipse (#2 and #3)
}

\begin{tikzpicture}
\draw \boundellipse{0,0}{5}{12};
\end{tikzpicture}
\end{document}

I can not use the above code to draw the ellipse with four vertices \left (-\dfrac{52}{5},-\dfrac{39}{5}\right ), \left (\dfrac{52}{5},\dfrac{39}{5}\right ), \left (-\dfrac{36}{5},\dfrac{48}{5}\right ), \left (\dfrac{36}{5},-\dfrac{48}{5}\right ). Semi-major-length of ellipse is 12 and semi-minor-length is 5.

How can I draw that ellipse.

Answer 1

this is rather extended comment ...

• this site is dedicated to tex problems
• as i can conclude you like to solve problem how to draw ellipse through four (arbitrary) vertices. this is math problem and is off-topic here

illustration how i understood your question:

mwe for above image:

\documentclass[border=3mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\usepackage{nccmath}

\begin{document}
    \begin{tikzpicture}[scale=0.5,% for scaling coordinates of vertices
E/.style args = {#1/#2}{ellipse, draw,
                 minimum width=#1cm, % major length
                 minimum height=#2cm,% minor length
                 inner sep=0pt},
   dot/.style = {circle, fill}
                    ]
\node[E=24/10, rotate=0] {};
\node (a) [dot,
    pin=225: {$\left(-\mfrac{52}{5},-\mfrac{39}{5}\right)$}] at (-52/5,-39/5) {};
\node (b) [dot,
    pin= 45: {$\left( \mfrac{52}{5}, \mfrac{39}{5}\right)$}] at ( 52/5, 39/5) {};
\node (c) [dot,
    pin=135: {$\left(-\mfrac{36}{5},-\mfrac{48}{5}\right)$}] at (-36/5, 48/5) {};
\node (d) [dot,
    pin=315: {$\left( \mfrac{36}{5},-\mfrac{48}{5}\right)$}] at ( 36/5,-48/5) {};
%
\draw[red, dashed] (a) -- (d) -- (b) -- (c) -- (a);
% elipse 1
\node[E=24/10] {};  % <major length>/<minor length>
% elpise 2
\node[E=24/10, rotate=-9, dashed] {};
\end{tikzpicture}
\end{document}

when you have correct determined coordinates of vertices and calculated ellipse rotation angle (if needed), than you ask, how to draw such ellipse. basic idea, how can be ellipse drawn, is shown in above mwe.