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Parabola.tex
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\documentclass[10pt]{article} \usepackage{amssymb,amsmath} \usepackage[hmargin=1cm,vmargin=0.5cm]{geometry} \begin{document} \noindent {\Large Parabola} \begin{align*} \text{In the following,}& \quad a\neq 0,\\ &\quad\text{$P(x_1,y_1)$ or $P(2ap,ap^2)$ is on the parabola, so is}\\ &\quad\text{$Q(x_2,y_2)$ or $Q(2aq,aq^2)$, and}\\ &\quad\text{$T(x_0,y_0)$ is the intersection of the}\\ &\qquad\qquad\text{two tangents from $P$ and $Q$.} \end{align*} % % The Basics % \begin{align*} \text{\large\bf The Basics}\qquad\qquad\qquad\qquad\qquad\qquad\\ \text{When focus is on x-axis:}\\ \text{Cartesian equation:}&\quad y^2=4ax\\ \text{Parametric equation:}&\quad y=2at,\quad x=at^2\\ \text{Eccentricity:}&\quad e=1\\ \text{x-intercept:}&\quad (0,0)\\ \text{Foci:}&\quad S=(a,0)\\ \text{Directrices:}&\quad m:\:x=-a\\ \text{Locus:}&\quad P(x,y)\::\quad PS=PL\:,\quad\text{where}\:PL \perp m\\ \\ \text{When focus is on y-axis:}\\ \text{Cartesian equation:}&\quad x^2=4ay\\ \text{Parametric equation:}&\quad x=2at,\quad y=at^2\\ \text{Eccentricity:}&\quad e=1\\ \text{y-intercept:}&\quad (0,0)\\ \text{Foci:}&\quad S=(0,a)\\ \text{Directrix:}&\quad m:\:y=-a\\ \text{Locus:}&\quad P(x,y)\::\quad PS=PL\:,\quad\text{where}\:PL \perp m \end{align*} % % Tangents and Chords (Cartesian) % \begin{align*} \text{\large{\bf Tangents and Chords} (Cartesian)}&\quad x^2=4ay\\ % \text{Derivative:}&\quad\frac{dy}{dx}=\frac{x}{2a}\\ \text{Tangent at $P$:}&\quad x_1 x=2a(y+y_1)\\ \text{Normal at $P$:}&\quad 2ax+x_1 y=2ax_1+x_1 y_1\\ \text{Intersection $T$:}&\quad(x_0,y_0)=\left(\frac{2a(y_2-y_1)}{x_2-x_1},\frac{x_1 y_2-x_2 y_1}{x_2-x_1}\right)\\ \text{Chord of Contact $PQ$:}&\quad x_0 x=2a(y+y_0)\\ \text{Focal Chord:}&\quad\text{$T$ is on directrix}\Leftrightarrow\text{$PQ$ is a focal chord}\\ &\text{\em Proof:}\:\:y_0=-a\Leftrightarrow S=(0,a)\:\text{satisfies}\:x_0 x=2a(y+y_0)\\ % % Tangents and Chords (Parametric) % \text{\large{\bf Tangents and Chords} (Parametric)}&\\ \text{Derivative:}&\quad\frac{dy}{dp}=2ap\:,\quad\frac{dx}{dp}=2a\:,\frac{dy}{dx}=p\\ \text{Tangent at $P$:}&\quad y=px-ap^2\\ \text{Normal at $P$:}&\quad x+py=2ap+ap^3\\ % \text{Intersection $T$:}&\quad\big(a(p+q),apq\big)\\ % \text{Chord of Contact $PQ$:}&\quad y=\frac{1}{2}(p+q)x-apq\\ \text{Focal Chord:}&\quad pq=-1\Leftrightarrow\text{$PQ$ is a focal chord}\\ &\text{\em Proof:}\:\:pq=-1\Leftrightarrow S=(0,a)\:\text{satisfies}\:y=\frac{1}{2}(p+q)x-apq \end{align*} % % Hyperbola approaching Parabola % \begin{align*} \text{\bf Parabola}\quad\text{\--- the case of a hyperbola with}&\quad e\to 1^{+}\\ \text{Hyperbola with foci on y-axis, shifted down by $b$:}&\quad\frac{(y+b)^2}{b^2}-\frac{x^2}{a^2}=1\\ \text{When $2kb=a^2$ ($k\neq 0$), upper half is}\:(y\geq 0):&\quad y=b\left(\sqrt{1+\frac{x^2}{a^2}}-1\right) =\frac{a^2}{2k}\left(\sqrt{1+\frac{x^2}{a^2}}-1\right)\\ &\quad\:\: =\frac{a^2}{2k}\left(\sqrt{1+\frac{x^2}{a^2}}-1\right)\cdot\frac{\sqrt{1+\frac{x^2}{a^2}}+1}{\sqrt{1+\frac{x^2}{a^2}}+1}\\ &\quad\:\: =\frac{a^2}{2k}\frac{\left(1+\frac{x^2}{a^2}\right)-1}{\sqrt{1+\frac{x^2}{a^2}}+1} =\frac{x^2}{2k}\frac{1}{\sqrt{1+\frac{x^2}{a^2}}+1}\\ \text{When $a\to+\infty$ :}&\quad y=\frac{x^2}{4k}\:,\quad\text{k becomes the focal length of the parabola.}\\ \text{Eccentricity:}&\quad \lim_{a\to\infty}e=\lim_{a\to\infty}\sqrt{1+\frac{a^2}{b^2}}=\lim_{a\to\infty}\sqrt{1+\frac{4k^2}{a^2}}=1\\ \end{align*} \end{document}