Two parabolas, perpendicular axes: Can we prove the intersection points are concyclic? In  it is shown that when a pair of parabolas have perpendicular axes and intersect at four points, the four points are concyclic (belong to the same circle).  It's easy to show that this is true in general, if we have coordinate equations in the form   $ax^2+bxy+cy^2+dx+ey+f=0, a'x^2+b'xy+c'y^2+d'x+e'y+f'=0$  for each parabola, by taking a linear combination of the two equations you can make the quadratic component a multiple of $x^2+y^2$ (from the hypothesis you can render $a'=c, b'=-b, c'=a$).  But is there a way to prove the claim by purely geometric techniques?  It smells like something that would have a purely geometric proof, but I can't see exactly how.
Intersection of two parabolae Problem: Consider two parabolae such that their axes of symmetry form a right angle. Prove that all four points of intersection lie on a common circle (it is an assumption that there exist such four points). Could you help me obtain a non-algebraic solution? (The algebraic one I did is given below.)  Algebraic solution:  There is a very simple algebraic approach, just introduce a Cartesian coordinate system such that the parabolae are given by equations:  $$y = \alpha_1 x^2 + \beta_1 $$ $$x = \alpha_2 y^2 + \beta_2 $$  and by scaling, adding and simplifying we can get the following ordinary circle equation (the right-hand side is positive because parabolae intersect): $$ \left(x-\frac{1}{2\alpha_2}\right)^2 + \left(y - \frac{1}{2\alpha_1}\right)^2 = \frac{-\beta_1}{\alpha_1} + \frac{-\beta_2}{\alpha_2} + \left(\frac{1}{2\alpha_1}\right)^2 + \left(\frac{1}{2\alpha_2}\right)^2$$  Non-algebraic solution: ??
Why xargs can't work with tail -f? I want tail -f /var/log/syslog | grep it with pattern "arpwatch" and send every line to myself via jabber: xmpp username@jabber.server using xargs  tail -f /var/log/syslog | grep arpwatch | xargs sendxmpp username@jabber.server   no working.  but tail  /var/log/syslog | grep arpwatch | sendxmpp username@jabber.server   works well.  I think it's some fundamental thing about xargs and tail -f that I don't understand. In Bash, suppose I visit a directory, and then another directory. I would like to copy a file from the first directory to the second directory, but without specifying the long pathnames of them. Is it possible?   My temporary solution is to use /tmp as a temporary place to store a copy of the file. cp myfile /tmp when I am in the first directory, and then cp /tmp/myfile . when I am in the second directory. But I may check if the file will overwrite anything in /tmp.  Is there something similar to a clipboard for copying and pasting a file? I came across this strange code snippet which compiles fine:  class Car {     public:     int speed; };  int main() {     int Car::*pSpeed = &amp;Car::speed;     return 0; }   Why does C++ have this pointer to a non-static data member of a class? What is the use of this strange pointer in real code? Get referenced node fields in twig Why do non-constant periodic functions have no limit at infinity? What is the fastest way to create a 90° joint? I often want to create joints like this:    Since I end up making these more than a little, I'd like a fast way to do it (i.e. on the order of &lt; 10 seconds)  I've found a couple tools which come close to being the ideal solution, but I've had some problems with them as listed below:   When using the spin tool I usually end up having to fiddle with the location for a while in order to to get a natural curve:    Are there any techniques to expedite this process of fiddling? The bridge tool is nice because I can precisely specify where both ends of the joint will be, however I often end up with subtly uneven joints, probably due to my initial mesh being not quite square.    Is there a fast way to correctly line up the initial edge loops in a way that will result in an even joint? Hello recently I went to best buy and played with a Nikon B500, it was great. I was able to focus on the smallest things, close or far, zoomed all the way in and all the way out. So I went on the internet and found out that the B700 shoots in raw so I got it instead thinking it was better. Only I cant get it to focus on anything close up, weather in macro or not after like 25% zoom it no longer focuses on anything no matter what settings I put it on leaving me to think that I should have gotten the cheaper B500. It may not shoot in raw but at least it would be usable. I am fairly new to photography I had a friend with much more experience come over and try to figure it out. He told me that my camera has a range that it is able to focus on. Only the cheaper B500 focused on things way more zoomed in and with the camera much closer to the object so it doesn't make much since to me. So my question is why wont my camera focus on macro shots (weather in macro or not)? I shall assume that the sun is at rest, so before the half mass disappear, the orbit of earth is an ellipse. This happens pretty much every time I try to bake normal maps. It's quite a broad error message and I still don't know exactly what the problems I'm having are, only that if I keep clicking things for half an hour eventually it sometimes just works.  My understanding of the things needed:   A "Generated" blender Image that the normals are written to (In UV/Image Editor, click Image->New)   Tell blender that this is the bake output, below (this is the really confusing bit which blender seems to try to guess and fails)  A UV Map for the low poly model (you only need the geometry of the high poly and it doesn't need a UV map) Under Bake, in the rendering tab:   Bake Mode = Normals Normal Space = Tangent Selected to Active = ✓ (for the next bit)  Select the high rez object first, then the low poly object With the two objects selected, click Bake (you may have to move the mouse to see the error pops up)   Now, from what I can tell the only  (with regard to the setup and apart from the internals of baking) is how blender finds the output image:   Cycles  Have a material using an ImageTexture node which points to your normal map output. Then  says the node must be selected, but  doesn't mention selecting. Blender Render  Have a texture attached to the material that uses the image.  is pretty vague on assigning the output image. Sometimes I've found hitting tab to go to edit mode and selecting the normal map image in the UV editor to help, or at least it seemed to.   I would like to know the exact details of what blender goes looking for when you hit Bake, rather than have to guess by following tutorials. For example, it seems to look for the high and low poly objects in your selection and based off the order you select them. How does it find the normal map/image to write to for both cycles and blender render? Does the Texture need to Influence Normals or does it just need to reference the Image? etc... which analysis fit the best for the data? I am trying to work on the data of my master thesis. I have to create a model, but I don't know which analysis perform. My data look like this     number ID sp ab alt SWdiv     1 A1 4 6 630 0.25802965   2 A2 9 16 750 0.29235324   3 A3 8 14 850 0.411620203   4 A4 9 22 950 0.444079553   5 A5 2 12 1050 0.294855765   6 A6 13 40 1130 0.298451014   7 B1 2 2 750 0.275329152   8 B2 5 6 850 0.043352118   9 B3 14 16 950 0   10 B4 17 58 1050 0.230358688   11 B5 7 9 1150 0.298759725   12 B6 8 68 1250 0.012192474   13 C1 12 20 830 0.467542842   14 C2 13 19 930 0.454480722   15 C3 6 30 1030 0.464584038   16 C4 7 14 1130 0.448191833   17 C5 5 15 1250 0.052045275   18 C6 2 7 1315 0.051767368   19 D1 3 3 650 0.294855541   20 D2 11 17 750 0.291717599   21 D3 9 46 850 0.084284864   22 D4 8 57 950 0.088574421   23 D5 4 32 1050 0.160873283   24 D6 13 22 1150 0.297638962   25 D7 6 22 1250 0.290733158   26 E1 11 23 800 0.250878932   27 E2 8 8 900 0.418021272   28 E3 10 13 1000 0.132159303   29 E4 5 10 1100 0.058487554   30 E5 5 15 1200 0.250310822   31 E6 10 19 1350 0.01600807   32 F1 6 20 800 0.462497319   33 F2 15 54 900 0.376552904   34 F3 6 13 1000 0.298264791   35 F4 6 13 1100 0.394736812   36 G1 14 22 830 0.391455956   37 G2 11 16 930 0.282922551   38 G3 5 10 1030 0.282398234   39 G4 10 15 1130 0.153502726   40 G5 13 18 1230 0.051586706   41 H1 10 11 700 0.3009609   42 H2 7 14 800 0.579228102   43 H3 5 11 900 0.528108703   44 H4 20 48 1000 0.28689587   45 H5 13 18 1100 0.264542772   46 H6 10 20 1250 0.29928462   47 I1 12 13 535 0.35326876   48 I2 3 5 635 0.273947657   49 I3 18 30 720 0.467208221   50 I4 16 188 835 0.276669711   51 I5 7 69 935 0.000867067   52 L1 6 10 435 0.36910895   53 L2 7 20 606 0.236700524   54 L3 14 33 697 0.111132519   55 L4 17 42 797 0.296757045   56 L5 5 7 910 0.27539552   57 L6 10 15 1000 0.29748451   58 L7 6 11 1101 0.276755556     58 sample points each has two value: altitude and diversity index related to habitat heterogeneity. What I sample are different insects and I have two values: sp = species (total number of different species for each sample point) and ab = abundance (total number of insects for each sample point). My question are are the species richness or the abundance affected by the altitude or the habitat heterogeneity? which is the best analysis to perform? and why? thank you for your help. How do I reverse a list using recursion in Python? I installed Linux Mint in 2 partitions: / and /home  Windows 8 is under UEFI boot mode.  How to hide those linux partitions in Explorer?    When I click on them: was closed as "Not constructive" a little while ago by CasperOne. I'm not here to argue about the closure; in fact, I'm forced to agree that, as it stands, the question is explicitly a polling question and eligible to be closed for that reason.  I'd like to see it preserved, however; it's got a fair bit of useful in-depth information for ObjC/Cocoa developers on the "singleton" topic.* It's also (possibly more importantly) a great duplicate target -- have a look at the list of linked questions. I want to try to stave off deletion, and hopefully get it reopened.** (I did vote for reopening, but it expired.)  My thought is to edit the question itself into a "constructive" form, but after spending some time pondering how to do so, I'm not crazy about any of the ideas that I've come up with. Essentially they all involve reworking the question to fit the answers:   Rewrite the question so that the accepted answer fits exactly. "I've got this singleton implementation, but I really want the creation to happen earlier in my app's execution. How can I make that happen?" This is artificial and (in my opinion) doesn't make a particularly good question. Make the question still general, but more concrete and focused on the implementation give in the body. "What are the pitfalls associated with this singleton creation procedure? Are there other features that my singleton should have?" This is still skirting the border of discussion->NC, and none of the answers really respond to this exactly. Remove the implementation from the question and just ask for the "best" implementation. This obviously requires coming up with stringent criteria for "best", and, while it would preserve the relationship between the Q and A's best, might not even move it out of NC territory.   I'm looking for suggestions of ways to re-write this so that it can stick around. Alternatively, if you think I should just be advocating for a historical lock, say so.   If you think my basic premise is faulty and don't think the question should be preserved at all, I respectfully request that you just downvote me instead of answering -- I'd prefer not to turn this into an argument about the merits of the question, and I definitely don't want it to become a(nother) "deletionist"/"inclusionist" skirmish (and I am not infrequently ).    *Quite a bit of it unfortunately in comments.  **Historical locking would also be an acceptable solution to me, but it doesn't seem to fit the "controversial" criterion. Usage of: 'since', 'for' and 'in' How to make a safe in vanilla Minecraft that locks and unlocks with a key? How can I make a safe/vault or something to lock my door that is very easy (as I am not that advanced with redstone) to keep people out of my house? Without mods? Does the Mordenkainen's Magnificent Mansion spell's mansion exist on a separate plane of existence? In particular: the  spell (PHB, p. 281) states:  [...] The creature can be anywhere on the same plane of existence as you. The spell ends if you or the target are no longer on the same plane. [...]  Can telepathy connect two characters, one inside the extradimensional dwelling created by the  spell and the other in the same plane as the portal to the magnificent mansion? How can I install Sun/Oracle's proprietary Java JDK 6/7/8 or JRE? I want to install Oracle's JRE and to update to the latest version with the Software Updater when they released. Is there a Ubuntu package that is provided by Canonical or Oracle?  Before release Java 7, I followed  to install Java 6.  But it doesn't work for Java 7. There is no package sun-java7-xxx. How can you install Java 7? How to prove $( A \times B ) \cap ( C \times D ) = ( A \cap C ) \times ( B \cap D )$ ? In this  to a question about a series, a theorem was stated:     If $A= \{a_i \}$ is a set such that $\sum_{i = 1}^{\infty} \frac{1}{a_i}$ converges, then $d(A) = 0$, where $d(A)$ is the  of the set.   My background in number theory is basically zero and all my attempts to prove this have been utterly unseccessful; would anyone please help me, or at least provide me with a hint to prove it? What is the difference between a linear and a circular polarizer?