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{\large\bf Notes on Conic Sections}

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\noindent
{\bf Geometrical Definition:}
\begin{itemize}
\item focus, directrix, eccentricity
\item parabola
\item ellipse
\item hyperbola
\item second geometric definition for ellipse and hyperbola
\end{itemize}


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\noindent
{\bf Special Properties:}
\begin{itemize}
\item parabola

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\item ellipse

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\item hyperbola

\end{itemize}

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\noindent
{\bf Equations for Conics in (Shifted) Standard Position:}
\begin{itemize}
\item parabola

\item ellipse

\item hyperbola

\item general equation of a conic in shifted standard position

\end{itemize}


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\noindent
{\bf Rotation of Axes:}
\begin{itemize}
\item rotating counterclockwise through angle $\phi$ 
(equations for rotation:
$x = X\cos(\phi) - Y\sin(\phi)$, $y = X\sin(\phi) + Y\cos(\phi)$)

\item The general second order polynomial equation is 
$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.

\item FACT: rotating counterclockwise through
$\displaystyle \phi =
\frac{1}{2}\cot^{-1}\left(\frac{A-C}{B}\right)$
eliminates the $xy$-term in the general equation.
\end{itemize}

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