Why only 2 tangents can be drawn to a hyperbola?
Why are there only two tangents to a hyperbola from a point, instead of four?
I am writing my thesis,  and i am in restructuring phase - sorting theorems where they should belong thematically. And it would be helpful to have all the proofs hidden so i am not laden with the additional information.  Is it possible to add something like hideproof to documentclass options to omit them when compiling the latex file. Can a quasiclassical electron wave packet in elliptic orbit be formed from bound hydrogen-like eigenstates? Position probability densities of eigenstates of hydrogen-like systems have axial symmetry, so that the wavefunction too much resembles the circular orbits in Bohr's model. I'd like to have a demonstration of correspondence principle, where an electron would be localized to look somewhat like a classical particle, and move in an elliptic (non-circular) orbit around a nucleus.  It seems though that if we try to make a wave packet too localized, then it'll disintegrate (get scattered by the nucleus) too fast to see anything resembling an elliptic orbit. OTOH, if we take it too spread out, it'll have to be quite far from the nucleus so as to avoid hitting it, and thus it'll have high total energy, which may appear to be above ionization threshold (at least partially), after which it'd be quite hard to analytically calculate evolution of the wave packet.  Thus my question is: is it possible to form a more or less localized wave packet, which would (on average) move in an obviously-elliptic orbit (major/minor axes ratio of 4:3 or higher), and only require bound states to fully represent it? If yes, then what properties (FWHM, apocenter, angular momentum etc.) should it have for this to be possible? What dried food can I take across the American border from Canada? How should ambiguous tags be dealt with? Is it possible to configure Windows 7 to use YYYY-MM-DD instead of DD/MM/YYYY as the date format? How? Where? pgfplots: prevent single plot from being listed in legend In order to fill the area between two complex curves I use tables and the vertcat command (see ).   Now I don't want the plotted area to appear in my legend, just the two curves. Is there any command to prevent the yellow area plot from being added to my legend?  this could be a minimal example:  \documentclass{article} \usepackage{pgfplots}  \begin{document} \begin{tikzpicture}   \begin{axis}[area style,axis on top]     \addplot+[mark=none,fill=yellow,draw=none] {0.1*x^2} \closedcycle;     \addplot[mark=none,draw=red,line legend] {0.1*x^2};     \addplot[mark=none,draw=blue,line legend] {0.0*x^2};     \legend{red curve,blue curve};     % \legend{\empty,red curve,blue curve};   \end{axis} \end{tikzpicture} \end{document}   With this legend{...}-command the last plot is skipped, an additional \empty doesn't help, too.  Sadly, drawing the first to plots together with fill=yellow,draw=red is not an option for me. Does anybody know any inside hacks? As far as I know, there is no command shown in the manual. Thanks in advance. Although work and heat do the same thing (increase or decrease the internal energy of the system), There is still a fundamental difference between them. For example, The way in which entropy is defined is a very good way to differentiate between work and heat. But, why is there such a distinction between the two things? Is it the limitation of Newtonian mechanics that it never accounted for something like heat which could also change the energy of the system? Is the word "Thermodynamical work" or "Hidden work" suitable for heat? How to make the BibTeX reference list visible in a LaTeX document Im very new at LaTeX and I would appreciate some help solving this problem. I have created a BibTeX file containing all references to a document. In this text file BibTeX items are inserted and no other text are written, for instance @article{...}. When I call for this bib file from the main.tex file I use the commands:  \bibliographystyle{plain} \bibliography{thebibliography}   Now to my question. All i get when I compile this main.tex is the title (bibliography) for this chapter, otherwise the textfield is blank. Is this error occurring because I should write anything in the bib file or have I missed any lines in the main.tex file. Do multi-use keys get discarded when finished with? As with many resident evil games, you sometimes needs to collect certain keys to open certain doors, and resident evil 7 has plenty of these around.  Some keys are clearly single use, and get left in the lock in which you use them. However, there are a number of keys that can be used on multiple doors. For example, the Scorpion Key.  I am concerned that I might end up carrying these around needlessly and using up valuable inventory space, so...  Do these multi-use keys get discarded when you use them on all the doors they apply to? Or do you get a warning saying the key is no longer needed? (like previous games are known to do) Should I get Enlightened badge? Today I've got 10 upvotes for an answer. Answer was a first answer in the question. I got an Nice Answer badge for this, but no Enlightened badge.  Shouldn't I get both?  is my answer :) Syntax error while defining an array as a property of a class Performance difference between dot notation versus method call in Objective-C Is there a good reason to use "printf" instead of "print" in java? How to disable Alt + ` (backtick/grave) My son has been playing Minecraft on our home computer for a couple of years now.  He just got an iPad Mini 2 for his birthday. Is it possible to transfer his game and his saves onto his iPad Mini 2? What is the accepted/most commonly used way to manipulate dynamic (with all dimensions not known until runtime) multi-dimensional arrays in C and/or C++.  I'm trying to find the cleanest way to accomplish what this Java code does:  public static void main(String[] args){  Scanner sc=new Scanner(System.in);  int rows=sc.nextInt();  int cols=sc.nextInt();  int[][] data=new int[rows][cols];  manipulate(data); }  public static void manipulate(int[][] data){    for(int i=0;i&lt;data.length;i++)    for(int j=0;j&lt;data[0].length.j++){          System.out.print(data[i][j]);           }     }   (reads from std_in just to clarify that dimensions aren't known until runtime).  Edit:I noticed that this question is pretty popular even though it's pretty old.  I don't actually agree with the top voted answer.   I think the best choice for C is to use a single-dimensional array as Guge said below "You can alloc rowscolssizeof(int) and access it by table[row*cols+col].".  There is a number of choices with C++, if you really like boost or stl then the answers below might be preferable, but the simplest and probably fastest choice is to use a single dimensional array as in C.  Another viable choice in C and C++ if you want the [][] syntax is lillq's answer down at the bottom is manually building the array with lots of malloc's. Is the force which throws a body on a rotating disk outwards centrifugal force? If some object is kept in a radial groove made on a disk and the disk is rotated with say constant angular velocity.   Now as we know that the object moves outwards ie away from the centre of the disk. I want to know which force is it which throws the object outwards.   As centrifugal force is a pseudo force and hence when the observer is in ground frame it shouldn't be present so which force exactly pulls it out. How to make a different font and color of text in labels? Is it possible to make a bold or italic, and change the color of individual words Label?  Example: the first line is bold in red text, font 12, second row 10 green italic font Quantum Field Theory without LSZ, how is it possible? Abstract, Interface, Classes OH MY - Question on Approach