I have the following three points:   A: One focus of an ellipse C: The furthest point from this focus (the far side of the major axis) D: Some other arbitrary point on the ellipse   From there, it's trivial to find:   E: The "mirror point" of D across the major axis of the ellipse   I know it should be simple trigonometry from here to find B, the second focus of the ellipse, but I can't pull it off for whatever reason. Can anyone help me out here?  I'm working in polar coordinates and have the known focus of the ellipse at the origin.  Edit: Added point "E", which one can trivially find in my scenario. I think my scenario is distinct from , since we've added the restriction that one point is necessarily as far as possible from the focus along the major axis.  Edit 2:  I think I solved it. The distance between the two foci is given by: $$ \frac{4r_Cr_D-4r_C^2}{2r_D+2r_Dcos(\theta)-4r_C} $$ Where theta is the (positive) angle between C and D. I used law of cosines and the ellipse property if anyone wants to check my work.
I would like to find all ellipses which contain 2 given points and has one focus at origin (zero). All in 2D plane.   There are several possible approaches but I'm not sure which is the best - both looks quite difficult to solve algebraically.   using polar equation relative to focus with $(R_1,\phi_1),(R_2,\phi_2)$, being coordinates of points $$ R_1 = \frac{a(1-e^2)}{1-ecos(\phi_1-\theta)} $$ $$ R_2 = \frac{a(1-e^2)}{1-ecos(\phi_2-\theta)} $$ then for given $\theta$ solve for semimajor axis $a$ and eccentricity $e$ using deffinition of elipse as a set of points of the same distance from both foci. Given 2 points of cartesian coordinates $(x_1,y_1),(x_2,y_2)$ and one focus in origin $(0,0)$. For each given distance parameter $L$ solve for coordinates of second focus $(x_f,y_f)$, $$ L = \sqrt{x_1^2 + y_2^2} + \sqrt{(x_1 - x_f)^2 + (y_1 - y_f)^2} $$ $$ L = \sqrt{x_2^2 + y_2^2} + \sqrt{(x_2 - x_f)^2 + (y_2 - y_f)^2} $$ I can also first rotate the coordinate system (or my input points) by given angle (which is my arbitrary parameter) and then use some simplified equation of ellipse which has major axis parallel to x-axis which has just 2 degrees of freedom. But even after this rotation I don't see much simplification of algebraic solution.    Or is there any better way?    The resulting equations are quite difficult to solve. I wonder if there is some trick to simplify it and make it more elegant.  Also, because I would later to implement it into computational code where it would be computed manytimes per second, I would prefer solution inf form which is fast to evaluate numerically ( prefferably without goniometric or any transcedental functions )    Why I'm interested ? - I need it for some optimalization of orbital transfers for spaceship orbiting the Sun. I asked also in physics.stackexchange, and people recomand me that I should rather go to math section.
How do simulacra behave? Cleric build to maximize melee damage and spell healing Mark questions to receive notifications and updates? Is there any way to fix the wavy lines that appear when photographing a striped dress?   It's not just in viewing online: when it was printed at wallet size, the wavy lines showed up. I have an Asus Sabertooth Z170 motherboard, with a Samsung 950 Pro m.2 NVMe drive on it. I've heard in the past that installing Windows 7 on an NVMe drive was a significant amount of work. Its now August 2016, is it still difficult? I've seen updates from Microsoft saying they've added support for NVMe in Windows 7.  Can I just download Windows 7 from Microsoft () and put it on a USB drive and install?  Is it harder than that? I have a 2 year old son and another baby on the way. I just recently found out I'm pregnant but not sure how far just yet. My son rammed into my stomach like he is always doing but this time it made me start having small pains in my stomach. I rushed to the bathroom to make sure there wasn't any bleeding.  I'm concerned his rough play could hurt the fetus if it continues. How should I put a stop to it? On Cesàro convergence: If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $ rewrite script to use wp_remote_get instead of file_get_contents_curl Electrostatics and tv screens The warlock's Thirsting Blade eldritch invocation cannot be normally used with the (non-warlock) class feature Extra Attack, since both feature explicitly say you attack twice instead of once. But it seems to me that you could use both features (if you have them) when using Two-Weapon Fighting.  Assume your character is wielding two light weapons, one of which is your pact weapon. You take the Attack action on your turn to attack twice with your non-pact weapon using your  class feature.  Then, given the Two-Weapon Fighting rules, you use your bonus action to attack once using your pact weapon this time.  The  rules (PHB p. 195) state:     When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand.   But since your have the Thirsting Blade invocation, and you used the Attack action, you get to attack twice with your pact weapon.  The Thirsting Blade  (PHB p. 111) says:     You can attack with your pact weapon twice, instead of once, whenever you take the Attack action on your turn.   Does this combo work within the rules, or is there any interaction I'm not aware of? Find out which process is locking a file or folder in Windows How can I find out which process is locking a file or folder in Windows?  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It returns this error:  Unknown column 'capita' in 'where clause'   I can however, run the query if I replace capita with gdp/population.  I don't want to do gdp/population &gt; .... How can I do capita &gt; ...? Or is this not possible? Subfields of finite fields We know that if a finite field $F$ has characteristic $p$ (prime), then $F$ has cardinality $p^r$ where $r = [F:\mathbb{F}_p]$.   I'm now trying to say something about the possible cardinalities of subfields of $F$. I can see that there is a subfield of cardinality $p^s$ for each $s$ that divides $r$, given by the fixed field of the group generated by $\phi^s$, where $\phi$ is the Frobenius automorphism.  Now suppose $K$ is a subfield of $F$. Then (since both are additive groups), Lagrange gives us that $|K|$ divides $|F|$, so $|K| = p^t$ for some $1 \leq t \leq r $ (alternatively, $K$ contains $\mathbb{F}_p$ and so is a vector space over $\mathbb{F}_p$ and is thus isomorphic to $\mathbb{F}_p^t$, where $t = [K:\mathbb{F}_p]$). By considering the multiplicative group of units of $K$ and $F$ respectively, we get that $ p^t - 1$ divides $p^r -1$. I want to make the leap to $t|r$, but I'm failing to see why this needs to be true. Any help would be appreciated. Modify a class definition's annotation string parameter at runtime Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\,\mathrm dx$. Evaluate the integral $$\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\, \mathrm dx.$$  How can i evaluate this one? Didn't find any clever substitute and integration by parts doesn't lead anywhere (I think).  Any guidelines please?