Tweet
Login
Mathematics Crystal
You may switch between
tex
and
pdf
by changing the end of the URL.
Home
About Us
Materials
Site Map
Questions and Answers
Skills
Topic Notes
HSC
Integration
Others
Tangent
UBC
UNSW
Calculus Advanced
Challenges
Complex Numbers
Conics
Differentiation
Integration
Linear Algebra
Mathematical Induction
Motion
Others
Polynomial Functions
Probability
Sequences and Series
Trigonometry
/
Topics /
Conics /
Hyperbola.tex
--Quick Links--
The Number Empire
Wolfram Mathematica online integrator
FooPlot
Calc Matthen
Walter Zorn
Quick Math
Lists of integrals
List of integrals of trigonometric functions
PDF
\documentclass[12pt]{article} \usepackage{amssymb,amsmath} \usepackage[hmargin=1cm,vmargin=1cm]{geometry} \begin{document} \noindent {\Large Hyperbola} \begin{align*} \text{In the following,}& \quad a>0, \quad b>0,\\ &\quad\text{$P(x_1,y_1)$ or $P(a\sec\theta_1,b\tan\theta_1)$ is on the hyperbola, so is}\\ &\quad\text{$Q(x_2,y_2)$ or $Q(a\sec\theta_2,b\tan\theta_2)$, and}\\ &\quad\text{$T(x_0,y_0)$ or $T(a\sec\theta_0,b\tan\theta_0)$ is the intersection of the}\\ &\qquad\qquad\text{two tangents from $P$ and $Q$.} \end{align*} % % Hyperbola (Basics) % \begin{align*} \text{\large\bf The Basics}\qquad\qquad\qquad\qquad\qquad\qquad\\ % \text{When foci are on x-axis:}\\ \text{Cartesian equation:}&\quad\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\\ \text{Parametric equation:}&\quad x=a\sec \theta,\quad y=b\tan \theta\\ \text{x-intercept:}&\quad A=(a,0),\quad A'=(-a,0)\\ \text{y-intercept:}&\quad\text{none}\\ \text{Eccentricity:}&\quad e=\sqrt{1+\frac{b^2}{a^2}}\:,\quad e>1\\ \text{Foci:}&\quad S=(ae,0),\quad S'=(-ae,0)\quad\text{(further from the origin than $a$)}\\ \text{Directrices:}&\quad m:\:x=\frac{\:a\:}{e},\quad m':\:x=-\frac{\:a\:}{e}\quad\text{(closer to the origin than $a$)}\\ \text{Locus:}&\quad P(x,y)\::\quad PS-PS'=\pm 2a\\ \text{Asymptotes:}&\quad \frac{\:x\:}{a}=\pm\frac{\:y\:}{b}\\ \\ \text{When foci are on y-axis:}\\ \text{Cartesian equation:}&\quad\frac{y^2}{b^2}-\frac{x^2}{a^2}=1\\ \text{Parametric equation:}&\quad y=b\sec \theta,\quad x=a\tan \theta\:,\quad\text{where $\theta$ starts on +y-axis}\\ \text{or}&\quad y=b\sec\left(\theta-\frac{\pi}{2}\right),\quad x=a\tan\left(\theta-\frac{\pi}{2}\right)\:,\\ \text{i.e.}&\quad y=b\csc\theta,\quad x=-a\cot\theta\:,\quad\text{where $\theta$ starts on +x-axis}\\ \text{x-intercept:}&\quad\text{none}\\ \text{y-intercept:}&\quad A=(0,b),\quad A'=(0,-b)\\ \text{Eccentricity:}&\quad e=\sqrt{1+\frac{a^2}{b^2}}\:,\quad e>1\\ \text{Foci:}&\quad S=(0,be),\quad S'=(0,-be)\quad\text{(further from the origin than $a$)}\\ \text{Directrices:}&\quad m:\:y=\frac{\:b\:}{e},\quad m':\:y=-\frac{\:b\:}{e}\quad\text{(closer to the origin than $a$)}\\ \text{Locus:}&\quad P(x,y)\::\quad PS-PS'=\pm 2b\\ \text{Asymptotes:}&\quad \frac{\:x\:}{a}=\pm\frac{\:y\:}{b}\\ \end{align*} % % Tangents and Chords (Cartesian) % \begin{align*} \text{\large{\bf Tangents and Chords} (Cartesian)}&\quad\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\\ % \text{Derivative:}&\quad\frac{dy}{dx}=\frac{b^2}{a^2}\frac{x}{y}\\ \text{Tangent at $P$:}&\quad\frac{x_1 x}{a^2}-\frac{y_1 y}{b^2}=1\\ \text{Normal at $P$:}&\quad\frac{a^2 x}{x_1}+\frac{b^2 y}{y_1}=a^2+b^2\\ \text{Intersection $T$:}&\quad(x_0,y_0)=\left(\frac{a^2(y_2-y_1)}{x_1 y_2-x_2 y_1},\frac{b^2(x_2-x_1)}{x_1 y_2-x_2 y_1}\right)\\ \text{Chord of Contact $PQ$:}&\quad\frac{x_0 x}{a^2}-\frac{y_0 y}{b^2}=1\\ \text{Focal Chord:}&\quad\text{$T$ is on directrix}\Leftrightarrow\text{$PQ$ is a focal chord}\\ &\text{\em Proof:}\:\:x_0=\pm\frac{\:a\:}{e}\Leftrightarrow(S,0)=(\pm ae,0)\:\text{satisfies}\:\frac{x_0 x}{a^2}-\frac{y_0 y}{b^2}=1\\ &\quad\text{as}\:\frac{\pm\frac{\:a\:}{e}\cdot\pm ae}{a^2}-\frac{0}{b^2}=1 \end{align*} % % Tangents and Chords (Parametric) % \begin{align*} \text{\large{\bf Tangents and Chords} (Parametric)}&\quad x=a\sec\theta,\:\:y=b\tan\theta\\ \text{Derivative:}&\quad\frac{dy}{dx}=\frac{b\sec\theta}{a\tan\theta}\\ \text{Tangent at $P$:}&\quad\frac{x\sec\theta_1}{a}-\frac{y\tan\theta_1}{b}=1\\ \text{Normal at $P$:}&\quad\frac{ax}{\sec\theta_1}+\frac{by}{\tan\theta_1}=a^2+b^2\\ \\ \text{Intersection $T$:}&\quad\left(\frac{a\cdot\cos\left(\frac{\theta_1-\theta_2}{2}\right)}{\cos\left(\frac{\theta_1+\theta_2}{2}\right)},\:\:\frac{b\cdot\sin\left(\frac{\theta_1+\theta_2}{2}\right)}{\cos\left(\frac{\theta_1+\theta_2}{2}\right)}\right)\\ &\quad\text{\tiny$=\left(\frac{a\sin(\theta_1-\theta_2)}{\sin\theta_1-\sin\theta_2},\:\frac{-b(\cos\theta_1-\cos\theta_2)}{\sin\theta_1-\sin\theta_2}\right)$}\\ \\ \text{Chord of Contact $PQ$:}&\quad\frac{\:x\:}{a}\cos\left(\tfrac{\theta_1-\theta_2}{2}\right)-\frac{\:y\:}{b}\sin\left(\tfrac{\theta_1+\theta_2}{2}\right)=\cos\left(\tfrac{\theta_1+\theta_2}{2}\right)\\ &\quad\text{\tiny$\frac{\:x\:}{a}\cdot\frac{\sin(\theta_1-\theta_2)}{\sin\theta_1-\sin\theta_2}+\frac{\:y\:}{b}\cdot\frac{\cos\theta_1-\cos\theta_2}{\sin\theta_1-\sin\theta_2}=1$} \end{align*} \end{document}