If $m$ varies then find the range of $c$ for which the line $y=mx + c$ touches the parabola $y^2 = 8(x+2)$ .  My Attempt:  Put the value $y = mx + c$ in the parabola equation and then done $\Delta = 0$ or $\Delta &gt;0$  I am getting $16/(m^2 + 8m) &gt;0$  But in this, how do I neglect $m$?
if m varies then find the range of c for which the line $y=mx + c$ touches the parabola $y^2 = 8(x+2)$ .  I tried Put the value $y = mx + c in$ the parabola equation and then done $D = 0$ or $D&gt;0 $  I am getting $16/(m^2 + 8m) &gt;0 $  But in this , how do I neglect m
Let $A$ and $B$ be two real $n\times n$ matrices such that $AB-BA=A$ Prove that $A$ is not invertible and that $A$ is nilpotent.  My attempt is the following. It holds that $AB=(B+I)A$ If $A$ were invertible, $B=A^{-1}(B+I)A$. Taking trace on both sides yields $tr(B)=tr(B)+n$, hence $n=0$, which is a contradiction. I can't prove that $A$ is nilpotent though. How to loop the particle simulation indefinitely? For a given data matrix $A$ (with variables in columns and data points in rows), it seems like $A^TA$ plays an important role in statistics. For example, it is an important part of the analytical solution of ordinary least squares. Or, for PCA, its eigenvectors are the principal components of the data.  I understand how to calculate $A^TA$, but I was wondering if there's an intuitive interpretation of what this matrix represents, which leads to its important role? I have been denied a visa several times in a row. How does one deal with such serial refusals? Over the course of time here on TSE, I have observed individuals whose pattern of visa applications is best described as ''...     serial refusals - Two or more consecutive visa refusals issued by the same regime to the same person. Commonly for the same, or nearly the same, reasons. In some visa regimes Serial refusals can have a cumulative effect wherein the applicant's credibility is increasingly eroded, especially when the application pattern suggests a secondary agenda.   A common thread appears to be the need to visit the destination country (especially Schengen, the USA, or the UK) with a special sense of urgency. Also common in these cases are the following types of questions...   Should I just reapply once again? What are my chances? What can I do to improve my odds in the future? Can I somehow address the previous refusals to avoid them being a burden on all future applications? How can I break the pattern?      NB: This question is intended as a canonical about serial refusals. I have never been personally denied a visa. How do I bridge network interfaces? I have a laptop running 11.10 with 2 network interfaces, wired and wireless.  Wireless interface connects to the router with DHCP and connects to the internet. Wired interface is connected to a switch. I want the computers on the switch to get their IP addresses from the router not my laptop.     How would I go about setting this up? Bridging the interfaces? Where can I find legendary crafting materials? I am looking for the stuff above, so I can make non-duplicate proposals on area 51. I am trying to find a book (young adult) where the devil is the protaganist, and other gods appear as characters I believe it was written by a teenager, and might have a sequel.   It has been many years since I read it, but I believe it had the devil acting like a regular human. He wasn't evil. I think it incorporated a lot of (gods/characters) norse, roman, Christian, and pagan. How to get first N elements of a list in C#? I would like to use Linq to query a bus schedule in my project, so that at any time I can get the next 5 bus arrival times. How can I limit my query to the first 5 results?  More generally, how can I take a slice of a list in C#? (In Python I would use mylist[:5] to get the first 5 elements.) Show JDBC ResultSet in HTML in JSP page using MVC and DAO pattern How To: Execute command line in C#, get STD OUT results $\forall a > 0$ $\sum_{n=1}^{\infty} f(na)$ is convergent. Prove that $\int_{0}^{\infty}f(x) dx$ is convergent. every eigenvalue of $T$ has only one corresponding eigenvector up to a scalar multiplication For a linear transformation $T$ on a vector space $V$ of dimension $n .$ Suppose it is given that for some vector $\mathbf{v},$ the vectors $\mathbf{v}, T(\mathbf{v}), T^{2}(\mathbf{v}), \ldots, T^{n-1}(\mathbf{v})$ are linearly independent, then is it true that every eigenvalue of $T$ has only one corresponding eigenvector up to a scalar multiplication. Description of varieties in $\mathbb{P}^2\times \mathbb{P}^1$ If $[x:y]$ are coordinates of $\mathbb{P}^1$ and $[X:Y:Z]$ are coordinates of $\mathbb{P}^2$, what do the following varieties look like?   $V(x^2X+y^2Y+xyZ)\subset \mathbb{P}^2\times \mathbb{P}^1$ $V(x^2X^2+y^2Y^2+xyZ^2)\subset \mathbb{P}^2\times \mathbb{P}^1$   As of now, I don't quite know how to visualize these varieties. A detailed geometric picture would be especially welcome. "Mutable variable is accessible from closure" in a function passed to Array.prototype.every Why does "ls -all" show time for some files but only year for others? If you issue ls -all command some files or directories have the year and some have the time? Why do some show the year while others show the time? Is the time representative of the time the file was created? Efficient way to check collision with potentially hundreds of objects I am trying to make a top down shooter that could potentially have hundreds of objects at once at one point. I need to check collision with projectiles, walls, and enemies. I am currently using ArrayLists which probably won't suffice later on and was wondering if someone had a good method or knows a good method for checking all of those objects at once. This is going to be a game for smartphones and is built in java. Any ideas would be great or if you could point me in some direction that would be fantastic.  Thanks in advance! I recently upgraded Windows 7 to Windows 10. The biggest issue I have now is that the Windows Search service no longer works, making the Start menu mostly unusable.  If I go to Indexing Options in the control panel I see that Indexing is not running and the list of paths to index is empty.  Going to advanced options, the current index path is empty, and the new path is empty too. This is true even if I try to use the "New Index" button.  Using the troubleshooter won't help since the service doesn't start. If I try to manually start the "Windows Search" service, I get the error  "Error 3: Unable to find specified path" (this is the rough translation, the original message is in my language).  What can I do to find out which path/file is missing ? I tried to check with Process Monitor, but the only file (beside system libraries) the SearchIndexer.exe try to access seem to be c:\ProgramData\Microsoft\Search\Data\Application (everytime the service is started the whole folder c:\ProgramData\Microsoft\Search is deleted and recreated) and C:\Windows\System32\config\systemprofile (which exists).  Edit: all available updates are installed, and issuing      SFC /scannow   Doesn't reveal any issue I am new to blender (yes, laugh at the obligatory doughnut) and I when I set up the camera (I use NUM 0 to access the camera and move it into position with &quot;Lock camera to view&quot;) in the viewport it shows as the first attached image, but when I render it the image is further back as shown in the second image. Is this normal behavior or am I missing something? So I am in an introductory linear algebra(with proofs) class, and I came across the following theorem:  If $S$ is a subspace of a vector space $V(F)$ and is generated by $\{a_1,...,a_n\}$ then any subset $\{b_1,...,b_m\}$ of $S$ with $m&gt;n$ is linearly dependent.   The proof in the textbook is done through induction (as most proofs regarding finite sets are) but I really do not follow it and I feel there is a simpler way of doing it. A hint would be appreciated.