Drawing ellipse that passes through specified coordinates

Question

I'm trying to draw an ellipse that must pass through (0, -1), (0, 1), (-1, 0), (1, 0), (1, -1) and (-1, 1). My MWE is below. I know that, I'm missing some math here.

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}

\begin{document}

\begin{tikzpicture}

\draw [help lines] (-2, -2) grid (2, 2);
\draw[rotate=45] (0, 0) ellipse (0.5cm and 1.5cm);

\filldraw[black] (0, 0) circle(1.5pt);
\filldraw[black] (0, 1) circle(1.5pt);
\filldraw[black] (1, 0) circle(1.5pt);
\filldraw[black] (-1, 0) circle(1.5pt);
\filldraw[black] (0, -1) circle(1.5pt);
\filldraw[black] (1, -1) circle(1.5pt);
\filldraw[black] (-1, 1) circle(1.5pt);

\draw[<->] (-2.0, 0) -- (2.0, 0) node[right]{\footnotesize $X_{1}$};
\draw[<->] (0, -1.5) -- (0, 1.5) node[above]{\footnotesize $X_{2}$};

\node[above right] at (0, 1)   {\tiny $\left(0, 1\right)$}; 
\node[above right] at (1, 0)   {\tiny $\left(1, 0\right)$}; 
\node[above left]   at (-1, 0)  {\tiny $\left(-1, 0\right)$}; 
\node[below right] at (0, -1)  {\tiny $\left(0, -1\right)$}; 
\node[below right] at (1, -1)  {\tiny $\left(1, -1\right)$}; 
\node[above left]   at (-1,  1)  {\tiny $\left(-1, 1\right)$}; 

\end{tikzpicture}

\end{document}

Edit

The equation of the ellipse is $X_1^2 + X_1 X_2 + X_2^2 = 1$.

Answer 1

Output

You can manually perform the computations as suggested by JoeDub. You can then change

\draw[rotate=45] (0, 0) ellipse (0.5cm and 1.5cm);

into

\draw[rotate=45] (0, 0) ellipse (0.82cm and 1.4cm);

Code

%http://tex.stackexchange.com/questions/99714/drawing-ellipse-that-passes-through-specified-coordinates#99714
\documentclass[border=5,convert={density=150}]{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}

\begin{document}

\begin{tikzpicture}

\draw [help lines] (-2, -2) grid (2, 2);
\draw[rotate=45,color=blue,very thick] (0, 0) ellipse (0.82cm and 1.4cm);

\foreach \x/\y in {0/0, 0/1, 1/0, -1/0, 0/-1, 1/-1, -1/1}
\filldraw[black] (\x, \y) circle(1.5pt);

\draw[<->] (-2.0, 0) -- (2.0, 0) node[right]{\footnotesize $X_{1}$};
\draw[<->] (0, -1.5) -- (0, 1.5) node[above]{\footnotesize $X_{2}$};

\foreach \x/\y/\direction in {0/1/above right, 1/0/above right, -1/0/above left, 0/-1/below left, 1/-1/below right, -1/1/above left}
\node [\direction] at (\x,\y) {\tiny $\left(\x, \y\right)$}; 
\end{tikzpicture}

\end{document}

---

You can also do this without rotation by specifying your major axis and minor axis (first axis and second axis in the manual) through the use of the \pgfpathellipse command. Here it is with your MWE.

\documentclass[border=5,convert={density=150}]{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}

\begin{document}

\begin{tikzpicture}

\draw [help lines] (-2, -2) grid (2, 2);
\pgfpathellipse{\pgfpointorigin}{\pgfpointxy{1}{-1}}{\pgfpointxy{0.58}{0.58}} % Create an ellipse with one end of major axis passing through (1,-1) and with one end of minor axis at (0.58,0.58)
\pgfusepath{stroke}

\foreach \x/\y in {0/0, 0/1, 1/0, -1/0, 0/-1, 1/-1, -1/1}
\filldraw[black] (\x, \y) circle(1.5pt);

\draw[<->] (-2.0, 0) -- (2.0, 0) node[right]{\footnotesize $X_{1}$};
\draw[<->] (0, -1.5) -- (0, 1.5) node[above]{\footnotesize $X_{2}$};

\foreach \x/\y/\direction in {0/1/above right, 1/0/above right, -1/0/above left, 0/-1/below left, 1/-1/below right, -1/1/above left}
\node [\direction] at (\x,\y) {\tiny $\left(\x, \y\right)$}; 
\end{tikzpicture}

\end{document}

Note here that the length of the semi-minor axis is approximately the square root of 0.58^2 + 0.58^2.

Edit

I have used \foreach in the above codes to make them shorter.