Is there synthetic proof of the equation of hyperbola using this definition: A hyperbola is the set of all points in a plane such that the difference of the distances from two fixed points (foci) is constant?
Is there geometric or visual (but rigorous) proof for the equation of hyperbola whose foci are on the $x$-axis using only the traditional definition of hyperbola: a hyperbola is a set of points such that the difference of distances from each point to focus is constant $(|r_1 - r_2| = 2a)$?  Also, can you maybe geometrically explain asymptotes of hyperbola (why do they exist)?
In the Avengers movie, why is Hulk's behaviour inconsistent? In the Avengers movie, when      Bruce Banner is on the helicarrier, he becomes angry and transforms into the Hulk. Black Widow happens to be next to him and even though both are on same team, he attacks her and then Thor.  He becomes hostile to everyone on the ship.  But later on, in the fight against Chitauri troops, Hulk is very comfortably self-controlled.    Why is that?    The only thing I could think of is that it has been many days (a year?) since the Hulk had gotten angry, so he couldn't control himself.    But that's lame to be so puny when Bruce Banner was compared to Stephen Hawking in intelligence. How do you mark a struct template as friend? How to use Alt+Shift to switch keyboard layouts? I recently upgraded to Ubuntu 13.10, with the new keyboard handling system. The little "switch keyboard" option does not let me set it to Alt+Shift. How can I get this keyboard binding, as I have absolutely no interest in using the Mac-esque Super+Space combo? How does DES decryption work? Is it the same as encryption or the reverse? If DES decryption is the same as encryption done in reverse order, then how can the reversed S-Box convert 4 bits into 6 bits? There are four options. Which of them is correct?- (a)${p\over k}$ (b)${k\over p}$ (c)$\log({p\over k})$ (d)${\log p\over k}$ I somehow want to use the rule for evaluating this kind of sum using integration, i.e. $\int_{0}^{1} f(x) \ dx=\lim_{n\to\infty} {1\over n}\sum_{r=1}^{n}f({r\over n})$  But the given expression $\lim_{n\to\infty} [{1\over kn}+{1\over k(n+1)}+{1\over k(n+2)}+\cdots+{1\over k(n+p-k)}]={1\over k}\lim_{n\to\infty}{1\over n}\sum_{r=0}^{p-k}\frac{1}{1+{r\over n}}$, sum running from $0$ to $p-k$ instead of $n$. So, how to evaluate this limit? Thanks for assistance in advance. Calculating Statistics Based on Expression in QGIS? Is it possible to calculate the sum of polygon areas based on whether or not it has a value in another column?   For example:  In column 2 I have the area of polygons. I want to calculate the area of all polygons that contain 'AG' in column 1 or all polygons that are IN column 1. But calculate them separately so I can get the total area of each type in column 1. I was trying to suggest a retag of  on Seasoned Advice, but the system won't let me; I get the familiar message     Edits must be at least 6 characters; is there something else to improve in this post?   In general, that's not true for retagging; if you don't change the question body at all, it should work. I suspect the logic checking the edit is thrown off guard because of the title warning:     The question you're asking appears subjective and is likely to be closed       I found , but I cannot find any evidence (in SEDE) for invisible characters being present in the question's source. Also, that's supposed to be . I'm looking for some way to run a batch file (.bat) without anything visible to the user (no window, no taskbar name, .etc..).  I don't want to use some program to do that, I'm looking for something cleaner. I've found  that uses VBScript, but I don't really like using VBS, either. Calculating centroids of polygon using ArcPy? I am trying to calculate the centroids of a polygon and add the x- and y-coordinate to their respective fields:  arcpy.env.workspace = r"C:/..."  input_fc = "shapefiles/precincts_v.shp" arcpy.AddField_management(input_fc, "X", "DOUBLE") arcpy.AddField_management(input_fc, "Y", "DOUBLE")  arcpy.CalculateField_management(input_fc, "X", "!SHAPE.CENTROID@DECIMALDEGREES!.split()[0]", "PYTHON")  arcpy.CalculateField_management(input_fc, "Y", "!SHAPE.CENTROID@DECIMALDEGREES!.split()[1]", "PYTHON")    but I an invalid syntax error. Where is the mistake?  Edit: exact error message  Traceback (most recent call last):   File "C:\Users\mquentel\Dropbox\PLSS\CA voting data\Python\CONSTRUCT_merge_voting_and_census.py", line 17, in &lt;module&gt;     arcpy.CalculateField_management(input_fc, "X", "!SHAPE.CENTROID@DECIMALDEGREES!.split()[0]", "PYTHON")   File "C:\Program Files (x86)\ArcGIS\Desktop10.4\ArcPy\arcpy\management.py", line 3360, in CalculateField     raise e ExecuteError: ERROR 000539: SyntaxError: invalid syntax (&lt;expression&gt;, line 1) Failed to execute (CalculateField). work done is zero if displacement of point of application of force is zero but why can't we assume forces on COM and its displacement What is wrong with the De Broglie–Bohm theory a.k.a "Causal Interpretation" of quantum theory? Possible Duplicate:          I've heard of De Broglie–Bohm theory a.k.a causal interpretation of quantum theory. The predictions match accurately with with the nondeterministic quantum theory.  As a philosophy buff this one seems just like the classical universe. No funny new-age religion style gimmicks.  So why don't people use this interpretation instead of the Copenhagen "magic" interpretation?  P.S. I'm not an expert. I possess only superficial knowledge of quantum physics. Error: Declaration of MyClass::start_lvl() should be compatible with that of Walker_Nav_Menu::start_lvl() What changes after peer review but prior to final submission are acceptable for a conference paper? In several different contexts we invoke the  to justify whatever statistical method we want to adopt (e.g., approximate the binomial distribution by a normal distribution). I understand the technical details as to why the theorem is true but it just now occurred to me that I do not really understand the intuition behind the central limit theorem.  So, what is the intuition behind the central limit theorem?   Layman explanations would be ideal. If some technical detail is needed please assume that I understand the concepts of a pdf, cdf, random variable etc but have no knowledge of convergence concepts, characteristic functions or anything to do with measure theory. Why do I get a black rectangle instead of my external picture? I write some of my network notes in TeX. I wanted to make a table, and inside it insert an image:   .   I wrote a document:  \documentclass{article} \usepackage[demo]{graphicx} \usepackage{booktabs} \usepackage{array}  \begin{document} \begin{tabular}{m{3cm}m{3cm}m{5cm}} \toprule Configuration for ... &amp; Commands &amp; Topology\\ \midrule PPP: CHAP, PAP &amp; \#sh int se 0/0/0  &amp; \includegraphics[width=4cm, height=4cm]{chappap.png}\\  &amp; &amp; \\ \bottomrule \end{tabular} \end{document}   But instead of my network topology image, I see only this: .   I don't know what is wrong, cause my image is 100% in the same dir, as my *.tex file.  EDIT:  Changed my code as @Jubobs suggested, now everything is as I wanted:  \documentclass{article}     \usepackage{graphicx}     \usepackage{booktabs}     \usepackage{array}      \begin{document}     \begin{tabular}{m{3cm}m{3cm}m{5cm}}     \toprule     Configuration for ... &amp; Commands &amp; Topology\\     \midrule     PPP: CHAP, PAP &amp; \#sh int se 0/0/0  &amp; \includegraphics[width=4cm, height=4cm]{chappap.png}\\      &amp; &amp; \\     \bottomrule     \end{tabular}     \end{document} Prove $\sum_{i=1}^n i! \cdot i = (n+1)! - 1$? Prove the summation:  $$\sum_{i=1}^n i! \cdot i = (n+1)! - 1$$ using induction.   base case:  $n=1$: \begin{align*} \sum_{i=1}^1 i! \cdot i &amp;= (1+1)! - 1 \\ 1 &amp;= 2 - 1 \\ 1 &amp;= 1 \end{align*}   This is a question from my test review packet, currently have the base case completed and I am a bit lost on where to go from there. Any help/hints are appreciated. I can't change materials I was making a scene in blender and I wanted to change the texture from "Material_1" to "Material_2". It lets me change it but when I go to rendered view the part looks the same. can someone please help me bash: syntax error near unexpected token `}' I tried using this:  $ if [$a == 1] then { echo 'yes'; } fi;   but I get an error:  -bash: syntax error near unexpected token `}'   What is the correct format?  I tried several with no luck. How do you validate that two math expressions are equal? Let's say you have a few expressions like the following:  $$\begin{array}((x+17)^2 \\ x^2 + 34x + 289 \end{array} \\ 288 + \frac{x^2}{2} + \frac{x^2}{2} + 34x + 1 \\ [...] $$  You get the idea: there's an infinite number of ways of representing the same expression. The challenge is that my application is validating answers to math problems, input by math students practicing a certain concept. I'd like to make sure that whenever they submit an answer, the checker is completely insensitive to the issues of ordering, simplification, formatting etc. which would be plain frustrating. I'd also ideally like to support more than one variable such as x, y, z in the same expression.  What's a reliable way to check that two expressions are the same? The most brute force way I can think of would be to just plug a number into each variable of each expression and see what the result is.  I have a hunch however that there might be ways of getting false positives this way. That, and I might be either missing something really obvious about this, or there might be a better way of doing the comparison.  Would love to hear what you think. Are there cheap GPS or other navigator devices with downloadable maps for the whole world? I don't need a recommendation or a list but my knowledge of GPS and "navi"s and the differences between them is an utter mystery to me.  But often I'm lost in backstreets in random countries where access to something like a live Google Maps would be awesome. I absolutely cannot afford a smartphone with worldwide roaming though.  Can GPS's do what I want or am I barking up the wrong tree? If they can is it only high end ones than can?