Let $P(x,y,1)$ and $Q(x,y,z)$ lies on the curve $\displaystyle \frac{x^2}{9}+\frac{y^2}{4}=4$ and $\displaystyle \frac{x+2}{1}=\frac{\sqrt{3}-y}{\sqrt{3}}=\frac{z-1}{2}$ respectively. Then the square of minimum distance between $P$ and $Q$ is   process i try  parametric coordinate for ellipse is $\displaystyle x = 6\cos \theta,y=4\sin \theta$  and line is $x=\lambda-2,y=\sqrt{3}-\sqrt{3}\lambda,z=2\lambda+1$  how do i solve it help me
Let $P(x, y, 1)$ and $Q(x, y, z)$ lie on the curves $$\frac{x^2}{9}+\frac{y^2}{4}=4$$ and $$\frac{x+2}{1}=\frac{y-\sqrt{3}}{\sqrt{3}}=\frac{z-1}{2}$$ respectively. Then find the square of the minimum distance between $P$ and $Q$.  My Attempt is:  I tried to find minimum distance between the points $(-2,\sqrt{3})$ and $(6\cos \theta,4\sin \theta)$.
What are the fundamentals of implementing hook_menu()?  I would like to see the basics covered in a single question, to avoid having to answer the same similar but different questions over and over. I wanted to show that $X' / U^\perp \cong U'$, for $U$ being a closed subspace of the Banach space $X$.  Therefore I looked at $l: X' / U^\perp \cong U' , x' + U^\perp=[x']  \mapsto x'|_U$.  It is clear that this map is onto, as we just take a $u' \in U'$. Then I can use Hahn-Banach to get an appropriate $x'$ such that $x'|_U = u'$ and in particular $||x'||=||u'||$.  Now I need to show that is isometric: since we always have $\|[x']\|\le \|x'\|$ and by Hahn-Banach we have $\|x'\|=\|x'_{|U}\|$, this shows $\|[x′]\|\le \|x'_{|D}\|$ and it should be always true that $\forall y' \in U^\perp: \|x'-y'\| \ge \|x'_{|D}\|$, hence $\|[x']\|\ge \|x'_{|D}\|$. Is this reasoning correct? The isometric property follows. How did "When" become the customary answer to "say when"? When a waiter at a restaurant comes by with pepper or Parmesan cheese, he says, "say when" and starts putting it on your food.  Many people will say "OK" or "that's enough," but it seems that the customary answer is "when".  How did this become the customary answer?  Did everyone decide to make the exact same joke, until it became commonplace.  I thought it was funny when I was a kid, but even now as an adult, I still say it.  For evidence that this is indeed commonplace, see .  The answer "when" is used and is external to the joke of the strip. But even so, I'm curious as to just how commonplace this really is.  I see it a lot in the US.  Is it used elsewhere in the world?  (by the way, I'm new to asking questions on this site, and I have no idea what tags to put on this question) Programmatically delete one entry from one field on one entity What's the proper way to programmatically delete one specific field entry for a multi-entry field from one specific entity? (that's specific fields and specific entities, not field types or instances and entity types)  Here's some possibilities I've looked into which don't work:   Loading the entity with entity_load(), setting the field entry to =null or =array(), then saving with field_attach_update(). It's the closest I've found but it leaves a zombie null field entry in the database which shows up as a blank row on forms until the form is saved and hook_field_is_empty() can kick in (hook_field_is_empty() is configured such that =null or =array() will flag as empty). Loading the entity with entity_load(), unsetting the field entry by key, then saving with field_attach_update(). This seems to do nothing - it seems that Drupal interprets the entry not being there as a sign to not modify it. (strangely I also sometimes get maximum query time expiration timeouts when trying this approach) field_attach_delete() - this is much too blunt: it kills all fields for the entity field_purge_data() - better, but still too blunt: it kills all entries of the field, not specific entries   So to clarify, I've got code that finds (and loads) an entity which has a field that has multiple entries, and finds a specific entry on that field that needs to be removed. I want to remove that entry completely, without touching any other entries or any other fields on the entity. Evaluate $\int \cos(3x) \sin(2x) \, dx$. Evaluate the indefinite integral    \begin{align} \int \cos(3x) \sin(2x) \, dx \end{align}   Please see my work attempt as my answer below. I have a  to which I'd like to add JPEG and PNG images that I generate on the fly.  All the examples I've seen so far in the , specially in the  use ImageIcons.  I'm generating these images as byte arrays, and they are usually larger than the common icon they use in the examples, at 640x480.   Is there any (performance or other) problem in using the ImageIcon class to display an image that size in a JPanel? What's the usual way of doing it? How to add an image to a JPanel without using the ImageIcon class?   Edit: A more careful examination of the tutorials and the API shows that you cannot add an ImageIcon directly to a JPanel. Instead, they achieve the same effect by setting the image as an icon of a JLabel. This just doesn't feel right... True or False: there is a space $X$ such that $S^1$ is homeomorphic to $X\times X$ I had an exam this morning, one of the questions asked about the truth of the statement      There is a space $X$ such that $S^1$ is homeomorphic to $X\times X$.   I said that this was false and this was my reasoning... was I correct? (Note, an earlier part of the question asked me to prove $\pi_1(Y\times Z)=\pi_1(Y)\times\pi_1(Z)$, I took this as a hint.)  Suppose for the sake of contradiction that such an $X$ exists. Then $\Bbb{Z}\cong G\times G$ where $G\cong\pi_1(X)$. Let $\Phi$ be the isomorphism.  Let $\Phi(1)=(g_1,g_2)\in G\times G$. Then at least one of $g_1,g_2$ is nonzero, since $\Phi(0)=(0,0)$ (the identity on $G\times G$, and $\Phi$ is injective). Without loss of generality $g_1\ne 0$. Then which element $n\in\Bbb{Z}$ has $\Phi(n)=(kg_1,0)$? Since $$kg_1=\underbrace{g_1+\dots +g_1}_k\in G,$$ and $\Phi$ is surjective, such an $n$ should exist. But this is not possible since $\Phi(n)=(kg_1,0)$ implies (since $\Phi$ is a homomorphism) that $kg_1=ng_1$ and $ng_2=0$, and since $k$ was arbitrary and this need not hold if $k\ne 0$. How to deal with "java.lang.OutOfMemoryError: Java heap space" error? How to prove that r is a double root if and only if it is a root of a polynomial and of its derivative. How would you make a comma-separated string from a list of strings? Were these other people Chronicoms? We know that Enoch is a Chronicom, which is to say a sort of alien temporal observer.   When Enoch and his agents came and kidnapped       Coulson’s team,   for the purpose of sending them where they wanted; were these other agents also Chronicoms? I’m inclined to think they must have been (would Enoch have shared their advanced technology and goals, etc. with mere mortals), but on the other hand only one pod was mentioned. How to change ubuntu install from legacy to uefi Find a function value given 2 points Given $f(x)$, which is differentiable at every point such that: $f'(x) \ge -5$ for every $ x \in R$   $f(2) = -13$, $f(9) = -48$  Prove that:$ f(3) = -18$  Now it's quite obvious that $ f(3) = -18$ since that if we know that the lowest value of the derivative is -5, the difference between $f(2)$ and $f(9)$ is $-48-(-13)=-35$  that means that that in the area $[2,9]$, $f'(x)=-5$ right? that's why I told that $f(3) = -18$, but I feel that my proof is insufficient, probably because I'm missing some theorems or such. can someone help me structure this proof into something RIGHT :)?  Thanks a lot! Serial voteup someone you learn from skype 4.3 cannot connect to account on ubuntu 14.04 LTS I have being using Skype for a long time on my Ubuntu 14.04 LTS. From this morning on, Skype simple cannot connect to my account anymore.  That is my Skype version:     Skype 4.3.0.37   © 2014 Skype and/or Microsoft   My Ubuntu system:  LSB Version:    core-2.0-amd64:core-2.0-noarch:core-3.0-amd64:core-3.0-noarch:core-3.1-amd64:core-3.1-noarch:core-3.2-amd64:core-3.2-noarch:core-4.0-amd64:core-4.0-noarch:core-4.1-amd64:core-4.1-noarch:cxx-3.0-amd64:cxx-3.0-noarch:cxx-3.1-amd64:cxx-3.1-noarch:cxx-3.2-amd64:cxx-3.2-noarch:cxx-4.0-amd64:cxx-4.0-noarch:cxx-4.1-amd64:cxx-4.1-noarch:desktop-3.1-amd64:desktop-3.1-noarch:desktop-3.2-amd64:desktop-3.2-noarch:desktop-4.0-amd64:desktop-4.0-noarch:desktop-4.1-amd64:desktop-4.1-noarch:graphics-2.0-amd64:graphics-2.0-noarch:graphics-3.0-amd64:graphics-3.0-noarch:graphics-3.1-amd64:graphics-3.1-noarch:graphics-3.2-amd64:graphics-3.2-noarch:graphics-4.0-amd64:graphics-4.0-noarch:graphics-4.1-amd64:graphics-4.1-noarch:languages-3.2-amd64:languages-3.2-noarch:languages-4.0-amd64:languages-4.0-noarch:languages-4.1-amd64:languages-4.1-noarch:multimedia-3.2-amd64:multimedia-3.2-noarch:multimedia-4.0-amd64:multimedia-4.0-noarch:multimedia-4.1-amd64:multimedia-4.1-noarch:printing-3.2-amd64:printing-3.2-noarch:printing-4.0-amd64:printing-4.0-noarch:printing-4.1-amd64:printing-4.1-noarch:qt4-3.1-amd64:qt4-3.1-noarch:security-4.0-amd64:security-4.0-noarch:security-4.1-amd64:security-4.1-noarch Distributor ID: Ubuntu Description:    Ubuntu 14.04.3 LTS Release:    14.04 Codename:   trusty   Any suggestion why or how to fix it?  Thanks in advance. Dirac delta function as a limit of sinc function Unable to boot into Windows after installing Ubuntu, how to fix? Did my workflow run on item changed or item creation? What exactly is the difference between a parametric and non-parametric model? I am confused with the definition of non-parametric model after reading this link  and .   Originally I thought "parametric vs non-parametric" means if we have distribution assumptions on the model (similar to parametric or non-parametric hypothesis testing). But both of the resources claim "parametric vs non-parametric" can be determined by if number of parameters in the model is depending on number of rows in the data matrix.  For kernel density estimation (non-parametric) such a definition can be applied. But under this definition how can a neural network be a non-parametric model, as the number of parameters in the model is depending on the neural network structure and not on the number of rows in the data matrix?  What exactly is the difference between parametric and a non-parametric model? Hiding axis text in matplotlib plots