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Ellipse Basics.tex
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\documentclass[10pt]{article} \usepackage{amssymb,amsmath} \usepackage{graphicx} \usepackage{wrapfig} \usepackage{txfonts} \usepackage[hmargin=2cm,vmargin=2cm]{geometry} \parindent0em \setlength{\parskip}{0.3cm} \begin{document} \hspace*{-0.5cm}{\large Ellipse Basics} \hspace*{-0.5cm}{\bf Foci and Directrices} An ellipse centred at (0,0) with its long axis along the $x$-axis and its short axis along the $y$-axis has the following equation: $\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$\quad where $a>b$. Let the $x$-intercepts of the ellipse be $A=(a,0)$ and $A'=(-a,0)$, the right focus $S=(k,0)$ and the right directrix $x=c$ intersecting the $x$-axis at $L=(c,0)$. The eccentricity $e$ of the ellipse is the constant ratio of $SP:PM$ where $PM\perp x=c$ (the directrix) for any $P$ on the ellipse. For point $A$, $e=\dfrac{SA}{AM_A}=\dfrac{a-k}{c-a},\quad a-k=ec-ae\quad\ldots(1)$. For point $A'$, $e=\dfrac{SA'}{A'M_{A'}}=\dfrac{a+k}{c+a},\quad a+k=ec+ae\quad\ldots(2)$. $(1)+(2):\quad 2a=2ec,\quad c=\dfrac{a}{e}.$\quad Similarly for the left directrix. So the directrices are\quad$\boxed{x=\pm\dfrac{a}{e}}$. $(2)-(1):\quad 2k=2ae,\quad k=ae.$\quad Similarly for the left focus. So the foci are\quad$\boxed{S=(ae,0),~~S'=(-ae,0)}$. \bigskip For a point $P$ on the ellipse directly above $S$, $x=k$ and $y =\sqrt{b^2\left(1-\dfrac{k^2}{a^2}\right)} =\sqrt{b^2\left(1-\dfrac{a^2e^2}{a^2}\right)} =\sqrt{b^2\left(1-e^2\right)} $~~. $e=\dfrac{SP}{PM} =\dfrac{y}{c-k} =\dfrac{\sqrt{b^2\left(1-e^2\right)}}{\frac{a}{e}-ae},\quad a-ae^2=\sqrt{b^2\left(1-e^2\right)},\quad a^2(1-e^2)^2=b^2(1-e^2),\quad 1-e^2=\dfrac{b^2}{a^2},\quad e^2=1-\dfrac{b^2}{a^2}>0. $ $\boxed{e=\sqrt{1-\dfrac{b^2}{a^2}}~.\quad 0