Given an ellipse $x^2/a^2+y^2/b^2=1,$ where $a\not=b,$ find the equation of the set of all points from which there are two tangents to the curve whose slopes are (a) reciprocals and (b) negative reciprocals.   First I let $P(c,d)$ outside the ellipse,then assume the linear equation: $y=(x-c)+d;$ second, according to the problem, the line must touch the ellipse, so substitute $y$ for the equation of ellipse to get the intersection, then let $D(\text{discriminant})=0.$ After all of this, I have no idea how to continue.
For a given ellipse, find the locus of all points P for which the two tangents are perpendicular.   I have a trigonometric proof that the locus is a circle, but I'd like a pure (synthetic) geometry proof.
Using Windows 10, I can't run certain commands from PowerShell but they all work correctly using cmd. PowerShell: node --version: node : The term 'node' is not recognized as the name of a cmdlet, function, script file, or operable program. Check the spelling of the name, or if a path was included, verify that the path is correct and try again. At line:1 char:1 + node --version + ~~~~     + CategoryInfo          : ObjectNotFound: (node:String) [], CommandNotFoundException     + FullyQualifiedErrorId : CommandNotFoundException  node.exe --version: no output. node.exe: opens a new black console (node.exe application itself). Get-Host | Select-Object Version: Version ------- 5.1.19041.906  Get-Command node: The term 'node' is not recognized as the name of a cmdlet, function, script file, or operable program. Check the spelling of the name, or if a path was included, verify that the path is correct and try again. At line:1 char:1 + Get-Command node + ~~~~~~~~~~~~~~~~ + CategoryInfo : ObjectNotFound: (node:String) [Get-Command], CommandNotFoundException + FullyQualifiedErrorId : CommandNotFoundException,Microsoft.PowerShell.Commands.GetCommandCommand Get-Command node.exe: CommandType Name Version Source ----------- ---- ------- ------ Application node.exe 14.16.1.0 C:\Program Files\nodejs\node.exe $env:Path output has nodejs &amp; git &amp; other paths. cmd: everything works fine with any command. I have git and nodejs in my PATH env variable. What's wrong with PowerShell, what a strange behavior is this and how to fix it? center align equation in ‎\documentclass[11pt,twoside,fleqn]{book}‎ If $A+B=AB$ then $AB=BA$ I was doing the problem $$ A+B=AB\implies AB=BA. $$  $AB=BA$ means they're invertible, but I can't figure out how to show that $A+B=AB$ implies invertibility. So, I know that $\max(-X_{(1)},X_{(n)})$ is a sufficient statistic for the parameter $\theta$. But can I also say that $(X_{(1)},X_{(n)})$ are jointly sufficient for the parameter $\theta$ ?  In other words, can a single parameter have jointly sufficient statistics? Can not add image inside SVG via jQuery:  tag becomes Script to raise a single window to the front How to use a field value in Template.php How to add a PPA on a server? Chrome asks for password to unlock keyring on startup In Google Chrome, when I go to a login page, a window pops up asking to "Enter password for keyring 'default' to unlock". In most cases, whether I click Cancel or enter my password, the login form gets auto filled anyway.  How do I get rid of the popup? I want it to auto login each time, not ask for my system password. The dialog box never appears for any other apps. given that $f$ is an analytic function with real part $u$ is a polynomial in the variable of $x,y$, $z=x+iy$, we need to show $f$ is a polynomial in $z$, I am kind of puzzled to see the problem, first I would like to say:any hint please Hiding axis text in matplotlib plots How do Belgian fries differ from the French fries we are accustomed to eat in the US and Canada in such places as McDonald's, Wendy's, Burger King and so on? Is it the variety of potato, the preparation involved or something totally different? What oil do they use in Belgium?   I remember having them while I was visiting family in Belgium. The fries were somewhat crispy and thicker than over here. Question about the Euclidean ring definition I recently came across the following definition for a euclidean ring:  There exists a function $g:R\to\Bbb N_0$ with the following properties:  1.) $\forall x,y \in R$ with $ y \neq 0$ there exist $q,r \in R$ with $x = qy + r$ where $r=0$ or $g(r) &lt; g(y)$.  2.) $\forall x,y\in R\setminus\{0\}$ we have $g(xy)\ge g(x)$  What I don't understand, is the point of the second requirement. I have seen definitions where this is not a requirement. What does 2.) achieve?   Thanks very much EL expressions not evaluated in JSP I'm starting  and finally decided to learn how to write epsilon-delta proofs.  I have been working on chapter 5, number 3(ii). The problem, in essence, asks to prove that     $$\lim\limits_{x \to 1} \frac{1}{x} = 1.$$    Here's how I started my proof,     $$\left| f(x)-l \right|=\left| \frac{1}{x} - 1 \right| =\left| \frac{1}{x} \right| \left| x - 1\right| &lt; \epsilon \implies \left| x-1 \right| &lt; \epsilon |x|$$   I haven't made any further progress past this point. Is it possible to salvage this proof? Should I try an alternate approach? How to tell whether a field is required or not in Apex by describe fields Let $A=\begin{pmatrix}0&amp;5\\ 7&amp;4\end{pmatrix}$.      1) Write $A$ as a product of $4$ elementary matrices.      2) Write $A^{-1}$ as a product of $4$ elementary matrices.   My work. I have managed to find $A^{-1}$, which came out to be this:  \begin{pmatrix}-\frac{4}{35}&amp;\frac{1}{7}\\ \frac{1}{5}&amp;0\end{pmatrix}  However, I am struggling to figure out how I would split each of these matrices into $4$ elementary matrices. Any help? I honestly have no idea how to approach this question. I have done easier questions eg. show $gcd(a,b)lcm(a,b)=ab$ by letting $a=Alcm(a,b)$ and $b=Blcm(a,b)$ and then using the fact that $A$ and $B$ are coprime. However this approach only works in two variables. Any help with the question in the title would be greatly appreciated, as well as the thought process behind the answer so that I can approach similar problems. What Nikon lens for both macro and portrait? All three lenses are available at a local store near me:   85mm 1.8 at about 400USD  85mm 1.4 at about 1000USD 105mm 2.8 at about 1015 USD   I plan on doing macro and portrait photography and have the ff thoughts so far:   I can save a lot with the 85mm 1.8 and a site claims that AF-wise it is much faster than the 85mm 1.4 at twice the price 105mm has VR but I don't know if that would be of much use with portrait and macro as I usually don't use slow shutter speeds with such subjects 105mm has AF-S which means I can get better insect shots because of the silence( and the 105 mm range) I don't know why I should prefer the 85mm 1.4 over the 105mm, when the 85mm may only have AF speed as an advantage, though I am not so sure if it is faster   so as you can see, I'm leaning towards 105mm f/2.8 or 85mm f/1.8 and less towards the 85mm f/1.4. However, I don't want to finalize the decision as many also claims that the 85mm f/1.4 is so good.  Any professional insights to add in helping me make the decision? Proving $\sum_{k=0}^n(-1)^k\binom nk=0$ Show that $$\sum_{k=0}^n(-1)^k\binom nk=0$$ So for odd $n$ we have an even number of terms. So $\binom nk=\binom n{n-k}$ which have opposite signs. Thus the sum is 0.  For even $n$ we have that $$\sum_{k=0}^n(-1)^k\binom nk= \binom n0+\sum_{k=1}^{n-1}(-1)^k\binom nk+\binom nn$$ Now $$\sum_{k=1}^{n-1}(-1)^k\binom nk= \sum_{k=1}^{n-1}(-1)^k\left[\binom{n-1}k+\binom{n-1}{k-1}\right]$$ What would that sum be in the square brackets?