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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Conic Sections">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sat Mar 29 23:25:28 1997}
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\begin{document}


\section{Conic Sections\label{Conic sections}}

\subsubsection{Parabolas\label{parabolas}}

The graph of the equation $y=ax^{2}+bx+c$ is a U-shaped curve called a 
\textbf{parabola} that opens either upward or downward, depending on whether
the sign of $a$ is positive or negative. The lowest or highest point of the
parabola is called the \textbf{vertex} and the parabola is symmetric about
its \textbf{axis}.

\paragraph{Geometric Definition of a Parabola}

A \textbf{parabola} is the set of points in the place equidistant from a
fixed point $F$ (called the \textbf{focus}) and a fixed line $I$ (called the 
\textbf{directrix}).

\paragraph{Parabola with Vertical Axis}

The graph of the equation 
\[
x^{2}=4py 
\]

is a parabola with the following properties. 
\[
\text{%
\begin{tabular}{ll}
\textbf{Vertex} & $V(0,0)$ \\ 
\textbf{Focus} & $F(0,p)$ \\ 
\textbf{Directrix} & $y=-p$%
\end{tabular}
} 
\]

\begin{enumerate}
\item  The parabola opens upward if $p>0$.\FRAME{dtbpF}{138.0625pt}{94.4375pt%
}{0pt}{}{}{Figure }{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
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131.3125pt;original-height 89.125pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'par1.wmf';file-properties "XNPER";}}

\item  The parabola opens downward if $p<0$.\FRAME{dtbpF}{146.1875pt}{%
100.875pt}{0pt}{}{}{Figure }{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
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137.875pt;original-height 94.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'parabola2.wmf';file-properties "XNPER";}}
\end{enumerate}

\paragraph{\protect\vspace{1pt}Parabola with Horizontal Axis}

The graph of the equation 
\[
y^{2}=4px 
\]

is a parabola with the following properties. 
\[
\begin{tabular}{ll}
\textbf{Vertex} & $V(0,0)$ \\ 
\textbf{Focus} & $F(p,0)$ \\ 
\textbf{Directrix} & $x=-p$%
\end{tabular}
\]

\begin{enumerate}
\item  The parabola opens to the right if $p>0$.\FRAME{dtbpF}{143.125pt}{%
101.75pt}{0pt}{}{}{Figure }{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
143.125pt;height 101.75pt;depth 0pt;original-width 129.5pt;original-height
91.5pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'parabola3.wmf';file-properties "XNPER";}}

\item  The parabola opens to the left if $p<0$.\FRAME{dtbpF}{144.375pt}{%
99.875pt}{0pt}{}{}{Figure }{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
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87.3125pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'parabola4.wmf';file-properties "XNPER";}}
\end{enumerate}

\subsubsection{Ellipses\label{ellipses}}

An \textbf{ellipse} is an oval curve that looks like an elongated circle.

\paragraph{Geometric Definition of an Ellipse}

An \textbf{ellipse} is the set of all points in the plane, the sum of whose
distances from two fixed points $F_{1}$ and $F_{2}$ is a constant. These two
fixed points are the \textbf{foci} (plural of \textbf{focus}) of the ellipse.

\paragraph{Ellipse with Center at the Origin}

The graph of the equation 
\[
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\text{ or }\frac{x^{2}}{b^{2}}+%
\frac{y^{2}}{a^{2}}=1\text{ with }a>b>0 
\]

is an \textbf{ellipse} with center at the origin and having the following
properties.

\begin{enumerate}
\item  If the equation is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with $%
a>b>0$, the graph looks as follows:\FRAME{dtbpF}{161.8125pt}{114.375pt}{0pt}{%
}{}{Figure }{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
161.8125pt;height 114.375pt;depth 0pt;original-width
154.1875pt;original-height 108.375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'ellipse1.wmf';file-properties "XNPER";}}

\begin{enumerate}
\item  Vertices: $\left( \pm a,0\right) $

\item  Major axis: Horizontal, length $2a$

\item  Minor axis: Vertical, length $2b$

\item  Foci: $\left( \pm c,0\right) $, $c^{2}=a^{2}-b^{2}$
\end{enumerate}

\item  If the equation is $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$ with $%
a>b>0$, the graph looks as follows:\FRAME{dtbpF}{98.125pt}{116.25pt}{0pt}{}{%
}{Figure }{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
98.125pt;height 116.25pt;depth 0pt;original-width 92.8125pt;original-height
110.25pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'ellipse2.wmf';file-properties "XNPER";}}

\begin{enumerate}
\item  Vertices: $\left( 0,\pm a\right) $

\item  Major axis: Vertical, length $2a$

\item  Minor axis: Horizontal, length $2b$

\item  Foci: $\left( 0,\pm c\right) $, $c^{2}=a^{2}-b^{2}$
\end{enumerate}
\end{enumerate}

\paragraph{Definition of Eccentricity}

For the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ or $\frac{x^{2}}{%
b^{2}}+\frac{y^{2}}{a^{2}}=1$ (with $a>b>0$) the \textbf{eccentricity} $e$
is the number 
\[
e=\frac{c}{a} 
\]

where $c=\sqrt{a^{2}-b^{2}}$, the eccentricity of every ellipse satisfies $%
0<e<1.$

\subsubsection{Hyperbolas\label{hyperbolas}}

Although ellipses and hyperbolas have completely different shapes, their
definitions and equations are similar. Instead of using a \emph{sum} of
distances from two fixed foci, as in the case of an ellipse, use the \emph{%
difference} to define a hyperbola.

\paragraph{Geometric Definition of a Hyperbola}

A \textbf{hyperbola} is the set of all points in the plane, the difference
of whose distances from two fixed points $F_{1}$ and $F_{2}$ is a constant.
These two fixed points are the \textbf{foci} of the hyperbola.

\paragraph{\protect\vspace{1pt}Hyperbola with Center at the Origin}

The graph of the equation 
\[
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\text{ or }\frac{y^{2}}{a^{2}}-%
\frac{x^{2}}{b^{2}}=1\text{ with }a>0\text{, }\dot{b}>0 
\]

is a hyperbola with its center at the origin and with the following
properties.

\begin{enumerate}
\item  The equation is $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ with $a>0$%
, $\dot{b}>0$.\FRAME{dtbpF}{142.5pt}{111.875pt}{0pt}{}{}{Figure }{\special%
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0pt;original-width 135.5pt;original-height 105.9375pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'hyp1.wmf';file-properties
"XNPER";}}

\begin{enumerate}
\item  Vertices: $\left( \pm a,0\right) $

\item  Transverse axis: Horizontal, length $2a$

\item  Asymptotes: $y=\pm \frac{b}{a}x$

\item  Foci: $\left( \pm c,0\right) $,\qquad $c^{2}=a^{2}+b^{2}$
\end{enumerate}

\item  The equation is $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$ with $a>0$%
, $\dot{b}>0$.\FRAME{dtbpF}{138.0625pt}{92.5pt}{0pt}{}{}{Figure }{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 138.0625pt;height 92.5pt;depth
0pt;original-width 131.3125pt;original-height 87.3125pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'hyperbola2.wmf';file-properties
"XNPER";}}

\begin{enumerate}
\item  Vertices: $\left( 0,\pm a\right) $

\item  Transverse axis: Vertical, length $2a$

\item  Asymptotes: $y=\pm \frac{a}{b}x$

\item  Foci: $\left( 0,\pm c\right) $,\qquad $c^{2}=a^{2}+b^{2}$
\end{enumerate}
\end{enumerate}

\subsubsection{Shifting Conics\label{shifting conics}}

\paragraph{Shifting Graphs of Equations}

If $h$ and $k$ are positive real numbers, then replacing $x$ by $x-h$ or by $%
x+h$ and replacing $y$ by $y-k$ or by $y+k$ has the following effect(s) on
the graph in any equation in $x$ and $y$. 
\[
\begin{tabular}{clc}
\textbf{Replacement} &  & \textbf{How the graph is shifted} \\ 
\multicolumn{1}{l}{$x$ replaced by $x-h$} &  & $h$ units right \\ 
\multicolumn{1}{l}{$x$ replaced by $x+h$} &  & $h$ units left \\ 
\multicolumn{1}{l}{$y$ replaced by $y-k$} &  & $k$ units upward \\ 
\multicolumn{1}{l}{$y$ replaced by $y+k$} &  & $k$ units downward
\end{tabular}
\]

\paragraph{General Equation of a Conic Section}

The graph of the equation 
\[
Ax^{2}+By^{2}+Cx+Dy+E=0 
\]

where $A$ and $B$ are not both zero, is a conic or degenerate conic. In the
nondegenerate cases, the graph is

\begin{enumerate}
\item  a parabola if $A$ or $B$ is zero

\item  an ellipse if $A$ and $B$ have the same sign (or a circle if $A=B$)

\item  a hyperbola if $A$ and $B$ have opposite signs.
\end{enumerate}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Conic sections, general equation including rotations}{}{}{%
../../Help/DM3-6.tex#Conic sections}}

{\small \hyperref{Paper cup problem}{}{}{../../Help/DM7-8.tex#Paper cup
problem}}

{\small \hyperref{Plot 2D, rectangular}{}{}{../../Help/DM6-4.tex#Plot
expression}}

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