\documentclass[leqno,12pt]{article} \setlength{\oddsidemargin}{0.25in} \setlength{\evensidemargin}{0.25in} \setlength{\textwidth}{6in} \setlength{\textheight}{8.5in} \setlength{\headheight}{6ex} \setlength{\headsep}{4ex} \setlength{\topmargin}{-0.5in} \usepackage{amsthm} \usepackage{amssymb} \usepackage{amsmath} \usepackage{epsfig} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} %%%%%%%%% \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} %%%%%%%% \newtheorem{proposition}[theorem]{Proposition} %%%%%%%%% \newtheorem*{theorem*}{Theorem} %%%%%%%%%% \newtheorem*{lemma*}{The $d\delta$-Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem*{corollary*}{Corollary} \newtheorem*{remark*}{Remark} \newtheorem*{application*}{Application} %%%%%%%%%%%%theoremprime \gdef\theorext{} \def\lastheor{\addtocounter{theorem}{-1}} \renewcommand{\thetheorem}{\arabic{section}.\arabic{theorem}\theorext} \newenvironment{theoremprime}{\lastheor\gdef\theorext{$'$}}%% {\gdef\theorext{}} %%%%%%%%%from kac %%%%%%%%%%%%%%%%%%%%%%%%%%%%resize fonts for section headings \makeatletter \renewcommand\section{\@startsection {section}{1}{\z@}% {-3.5ex \@plus -1ex \@minus -.2ex}% {2.3ex \@plus.2ex}% {\normalfont\large\bfseries}} \renewcommand\subsection{\@startsection{subsection}{2}{\z@}% {-3.25ex\@plus -1ex \@minus -.2ex}% {1.5ex \@plus .2ex}% {\normalfont\normalsize\bfseries}} %\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% % {-3.25ex\@plus -1ex \@minus -.2ex}% % {1.5ex \@plus .2ex}% % {\normalfont\normalsize\bfseries}} \@addtoreset{equation}{subsection} \makeatother %%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% \numberwithin{equation}{section} \setcounter{section}{0} %\makeatletter %\@addtoreset{equation}{section} %\makeatother %\renewcommand{\theequation}{\arabic{equation}} \newcommand\Cinf{\mathcal{C}^\infty} %%%\newcommand{\exp}{\operatorname{exp}} \newcommand\A{\mathcal{A}} \newcommand\C{\mathcal{C}} \newcommand\F{\mathcal{F}} \newcommand\I{\mathcal{I}} \renewcommand\L{\mathcal{L}} \renewcommand\P{\mathcal{P}} \newcommand\Ss{\mathcal{S}} \newcommand{\CC}{{\mathbb C}} \newcommand{\LL}{{\mathbb L}} \newcommand{\RR}{{\mathbb R}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\Alt}{\mathop{\rm Alt\,}\nolimits} \newcommand{\Image}{\mathop{\rm Image\,}\nolimits} \newcommand{\Ker}{\mathop{\rm Ker\,}\nolimits} \newcommand{\red}{\mathop{\rm red\,}\nolimits} \newcommand{\sgn}{\mathop{\rm sgn\,}\nolimits} \newcommand{\supp}{\mathop{\rm supp\,}\nolimits} \newcommand{\vol}{\mathop{\rm vol\,}\nolimits} \newcommand{\st}[1]{\ensuremath{^{\scriptstyle \textrm{#1}}}} \newcommand{\alphaparenlist}{% changes enumerate 1st level to (a)...(z) \renewcommand{\theenumi}{\alph{enumi}}% \renewcommand{\labelenumi}{(\theenumi)}% } %\alphaparenlist \newcommand{\arabiclist}{% changes enumerate 1st level to 1...9 \renewcommand{\theenumi}{\arabic{enumi}}% \renewcommand{\labelenumi}{(\theenumi)}% } %\arabiclist \newcommand{\romanlist}{% changes enumerate 1st level to i...ix \renewcommand{\theenumi}{\roman{enumi}}% \renewcommand{\labelenumi}{(\theenumi)}% } \begin{document} \begin{center} \Large{\textbf{Notes for Geometry \\ Conic Sections}} \end{center} The notes is taken from {\it Geometry, by David A. Brannan, Matthew F. Esplen and Jeremy J. Gray, 2nd edition} \section{Conic Sections}\label{sec:1} A conic section is defined as the curve of intersection of a double cone with a plane. \vspace{0.3cm}\\ \underline{Figure:} \vspace{5cm}\\ \noindent \underline{Examples:} \begin{enumerate} \item \underline{non-degenerate conic sections}: parabolas, ellipses or hyperbolas; \item \underline{degenerate conic sections}: the single point, single line and pair of lines. \end{enumerate} \subsection{Focus-Directrix Definition of the Non-Degenerate Conics} The 3 non-degenerate conics can be defined as the set of points $P$ in the plane satisfying:\vspace{0.2 cm}\\ {\it The distance of $P$ from a fixed point $F$ (called the \underline{focus} of the conic) is a constant multiple (called its \underline{eccentricity}, e) of the distance of $P$ from a fixed line $d$ (called its \underline{directrix}).}\vspace{0.5 cm}\\ \underline{Eccentricity:} A non-degenerate conic is \begin{enumerate} \item an ellipse if $0 \le e < 1$, \item a parabola if $e=1$, \item a hyperbola if $e >1$. \end{enumerate} \vspace{0.5cm} \subsection*{\underline{Parabola ($e=1$):}} {\it A parabola is defined to be the set of points $P$ in the plane whose distance from a fixed point $F$ is equal to their distance from a fixed line $d$.} \vspace{2 ex}\\ \noindent We now derive a parabola {\it in standard form}: \vspace{2 ex}\\ Let $F(a, 0)$ be the focus and $d: x= -a$ be the directrix. Let $P(x,y)$ be an arbitrary point on the parabola and let $M(-a, y)$ be the foot of the perpendicular from $P$ to the directrix. \\ \underline{Figure:} \vspace{5cm}\\ \noindent Since $FP = PM$, by the definition of the parabola, this follows that $$ FP^2=PM^2 \Rightarrow (x-a)^2+y^2 = (x+a)^2 \Rightarrow y^2 =4ax. $$ The point $(at^2, 2at), t \in \mathbb{R}$ lies on the parabola. \[ \because (2at)^2 = 4a \cdot at^2. \] Conversely, we can write the coordinates of each point on the parabola in the form $(at^2, 2at)$. For if we choose $t = \frac{y}{2a}$, then $y=2at$ and $x = \frac{y^2}{4a}=\frac{(2at)^2}{4a}=at^2.$ It follows that there is a one-to-one correspondence between the real numbers $t$ and the points of the parabola. \vspace{1 ex} \noindent %\begin{tabular}{|l|}\hline {\bf Parabola in standard form} {\it A parabola in standard form has equation $$ y^2 = 4ax, \quad \text{where} \ a > 0.$$ It has focus $(a, 0)$ and directrix $x =-a$; and it can be described by the parametric equations: $$x=at^2, \quad y = 2at \quad (t \in \mathbb{R}).$$} %\\ \hline %\end{tabular} \noindent We call the $x$-axis the \underline{axis} of the parabola in standard form, since the parabola is symmetric with respect to this line. We call the origin the \underline{vertex} of a parabola in standard form, since it is the point of the intersection of the axis of the parabola with the parabola. A parabola has no center. \begin{example} Write down the focus, vertex, axis and directrix of the parabola $E$ with equation $y^2=2x.$ \end{example} \begin{proof}[Solution] Focus: $F=(\frac{1}{2}, 0)$, \ Axis: $x$-axis, \ Vertex: $(0, 0)$, \ Directrix: $x=-\frac{1}{2}$. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vspace{0.5cm} \subsection*{\underline{Ellipse ($0 \le e <1$):}} {\it We define an ellipse with eccentricity zero to be a circle. We define an ellipse with eccentricity $e$ (where $0 < e < 1$) to be the set of points $P$ in the plane whose distance from a fixed point $F$ is $e$ times their distance from a fixed line.} \vspace{2 ex}\\ \noindent We now derive an ellipse {\it in standard form}: \vspace{2 ex}\\ Let $F(ae, 0), a>0$ be the focus and $d: x = \frac{a}{e}$. Let $P(x,y)$ be an arbitrary point on the parabola and let $M(\frac{a}{e}, y)$ be the foot of the perpendicular from $P$ to the directrix. \\ \underline{Figure:} \vspace{5cm}\\ \noindent Since $FP = e \cdot PM$, by the definition of the ellipse, this follows that $$ FP^2=e^2 \cdot PM^2 \Rightarrow (x-ae)^2+y^2 = e^2 (x-\frac{a}{e})^2 \Rightarrow \frac{x^2}{a^2} + \frac{y^2}{a^2(1-e^2)} =1. $$ Let $b = a \sqrt{1-e^2}$. Then %$\because 0 < e < 1 \Rightarrow 0 < b < a.$ \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \] The equation is symmetrical in $x$ and $y$. The ellipse also has a second focus $F'(-ae, 0)$ and a second directrix $d': x= - \frac{a}{e}.$ We call the segment joining the points $(\pm a, 0)$ the \underline{major axis} of the ellipse and the segment joining the points $(0, \pm b)$ the \underline{minor axis} of the ellipse. $\because b < a$, the minor axis is shorter than the major axis. The origin is the \underline{center} of this ellipse. Note that $(a \cos t, b \sin t)$ lies on the ellipse. \[ \because \frac{(a\cos t)^2}{a^2} + \frac{(b \sin t)^2}{b^2} = \cos^2 t + \sin^2 t =1. \] We can check that $x=a \cos t, y=b \sin t, t \in (-\pi, \pi]$ gives a parametric representation of the ellipse. \vspace{1 ex} \noindent %\begin{tabular}{|l|}\hline {\bf Ellipse in standard form} {\it An ellipse in standard form has equation $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad \text{where} \ a \ge b > 0, \ b^2=a^2(1-e^2), \ 0 \le e < 1.$$ It can be described by the parametric equations \[ x =a \cos t, y= - b \sin t, \ t \in (-\pi, \pi]. \] If $e>0$, it has foci $(\pm ae, 0)$ and directrix $x = \pm \frac{a}{e}.$} %\\ \hline %\end{tabular} \vspace{0.2cm} \begin{exercise} \[ \sqrt{(x+ae)^2+y^2} + \sqrt{(x-ae)^2+y^2} =2a. \] \end{exercise} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vspace{0.5 cm} \subsection*{\underline{Hyperbola ($ e >1$):}} {\it A hyperbola is the set of points $P$ in the plane whose distance from a fixed point $F$ is $e$ times their distance from a fixed line $d$, where $e > 1.$} \vspace{2 ex}\\ \noindent We now derive a hyperbola {\it in standard form}: \vspace{2 ex}\\ Let $F(ae, 0), a>0$ be the focus and $d: x = \frac{a}{e}$. Let $P(x,y)$ be an arbitrary point on the parabola and let $M(\frac{a}{e}, y)$ be the foot of the perpendicular from $P$ to the directrix. \\ \underline{Figure:} \vspace{5cm}\\ \noindent Since $FP = e \cdot PM$, by the definition of the hyperbola, this follows that $$ FP^2=e^2 \cdot PM^2 \Rightarrow (x-ae)^2+y^2 = e^2 (x-\frac{a}{e})^2 \Rightarrow \frac{x^2}{a^2} - \frac{y^2}{a^2(e^2-1)} =1. $$ Let $b = a \sqrt{e^2-1}$. Then \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. \] The equation is symmetrical in $x$ and $y$. The ellipse also has a second focus $F'(-ae, 0)$ and a second directrix $d': x= - \frac{a}{e}.$ We call the segment joining the points $(\pm a, 0)$ the \underline{major (transverse) axis} of the hyperbola and the segment joining the points $(0, \pm b)$ the \underline{minor (conjugate) axis} of the hyperbola. The origin is the \underline{center} of this hyperbola. We can check that $x= \sec t, y = b \tan t, t \in (- \frac{\pi}{2}, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \frac{3\pi}{2})$ gives a parametric representation of the hyperbola. The hyperbola consist of $2$ branches. When $x \to \pm \infty$, the branches get closer and closer to the lines $y=\pm \frac{b}{a}x$ the asymptotes of the hyperbola. \vspace{1 ex} \noindent %\begin{tabular}{|l|}\hline {\bf Hyperbola in standard form} {\it A hyperbola in standard form has equation $$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \quad \text{where} \ a \ge b > 0, \ b^2=a^2(e^2-1), \ e > 1.$$ it has foci $(\pm ae, 0)$ and directrix $x = \pm \frac{a}{e}.$ It can be described by the parametric equations \[ x =a \sec t, y= - b \tan t, \ t \in (-\frac{\pi}{2}, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \frac{3 \pi}{2}). \] } \begin{example} Determine the foci $F$ and $F'$ of the hyperbola $E$ with equation $x^2-2y^2=1.$ \end{example} \begin{proof}[Solution] \[ \left( \pm \sqrt{3/2}, 0 \right). \] \end{proof} \vspace{0.2cm} \begin{exercise} \[ \sqrt{(x+ae)^2+y^2} - \sqrt{(x-ae)^2+y^2} = \pm 2a. \] \end{exercise} \end{document} % LocalWords: Munkres multivariable Spivak analogue Fredholm Plancherel