consider-the-ellipse-fracx2a2fracy2b21-where-half-the-length-of-the-major-axis-is-a-and-the-foci-are

Question

Consider the ellipse
$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$
where half the length of the major axis is $a$ , and the foci are $( \pm c, 0)$ such that $c^{2}=a^{2}-b^{2} .$ Let $L$ be the vertical line $x=a^{2} / c$
(a) Prove that L is a directrix for the ellipse. [Hint: Prove that PF/PD is the constant c/a, where P is a point on the ellipse, and D is the point on L such that PD is perpendicular to L.]
(b) Prove that the eccentricity is $e=c / a .$
(c) Prove that the distance from $F$ to $L$ is ale $-e a$

Mauricio Araiza Canizales · Accepted Answer

So we&#x27;re giving the following information about ah, on the lips This ellipse described by the equation X squared, divided by a squared plus y squared divided by the square A circle to one and we want to prove that the line xs equal to a square divided by sea is the directories for this ellipse to do so we&#x27;re gonna prove that the line segment P f which is this one right here, this one degraded by the line segment p d Just just this one is equal to the constant See over a to start off. We&#x27;re gonna take the following equation for an ellipse in polar question Polar Coordinates, which is art. A sequel to E. Tim&#x27;s game divided by one plus E Co Cynthia. And for an angle of fate icicle to zero, we&#x27;ll have the following a question. Our physical to eatem ski. You know I want plus e since the coastline of today&#x27;s just Europe and that is equal to a minus c. So remember that the distance from the center to this Burke takes over here is a the distance from the centre toe. One office folks, I it&#x27;s C and so the radius in this case, just gonna be the difference between A and C, which is this segment right here. This is our So now we&#x27;re just gonna sold for? We&#x27;re gonna take this equation and sold for dis constant right here. We&#x27;ll have e tem ski if you go to a minus C times one plus e And so we also know that the constant e sequel to see Divided by a So now I&#x27;m Minister working These new networks will have eight times. Kate is equal to a minus c tim&#x27;s one plus e well, to know that he is equal to see divided by a So now we&#x27;ll have the line segment pf just this one right here. P f That&#x27;s it for this line segment we have that the distance, the existence from the faux say to the point is just gonna be our co Cynthia. What? And so for this point right here, we&#x27;ll have that our radius over here just gonna be e temps que you ready for that one plus e co Cynthia Well, we have a stated that eatem ski. It&#x27;s just equal to e a minus C attempts one plus e. He&#x27;s just gonna beat the same. The denominators stays the same one plus eight times goes in theater. Well, we also have stated that he&#x27;s see divided by a we&#x27;ll have a minus C plus one plus c divided by a You better buy one plus see temps, eh? Cosign theater. And so now I&#x27;m gonna start working with gel giraffe over here. So we&#x27;ll have this same line segment, which is just gonna be a minus. C. I&#x27;m gonna distribute this term right here. I&#x27;m gonna multiply. Actually, I&#x27;m gonna multiply everything through so we&#x27;ll have a temps one just want a plus c divided by eight times eight is just see so negative C Times one is just negative c and so negative. See, divided by a It&#x27;s just negative. C squared, divided by a So now the denominator stays. I&#x27;m gonna multiply everything through buying a over here, someone a month supply everything might a on the top and on the bottom we&#x27;ll have one plus c divided by eight coast and failure. So we&#x27;ll have this age was gonna become a square. So these two see terms cancel each other minus C square. Since I have multiplied by in a term. So over here we&#x27;ll have a plus C co, Cynthia. And so now that we have this term break here, we have our we have a relationship or be square for our inner lips, which is B squared is equal to eight square minus C square. And this is exactly what we have over here. This is the same. So we can say that this term right here it&#x27;s gonna be be square divided by a ce temps cosign eight plus c times coasting theater. So we&#x27;re gonna step right here and we have the chairman on a question for the line segment pf now we&#x27;re gonna work our way through the segment slain segment P. D. So, in this case, P d just gonna be the distance between the X coordinate, which is X is equal to a square. Their embassy in our location for point D So the ex according it for a line Sigman Just gonna be a square. You ready, Casey? In our point, it&#x27;s just located from this from the origin. It&#x27;s located at a distance. See? Sorry C plus our cosa fada. So we have our distance from the center which is our point of reference. This is gonna be this sea part. Plus are our CO Cynthia, which depends on the angle at the point makes would respect to the folks I And so if we distribute this native sign will have a square divided by C minus C minus arco. Cynthia. And so this term we&#x27;re just going to say that we&#x27;re just gonna multiply everything through because we already have the chairman who have discs gonna be equal to a square minus C square. Yeah, but if I see I&#x27;m just I&#x27;m just multiplying this seat term times this one minus So we have determined that our in this case, which is the point, which is this point right here it&#x27;s also equal to our It&#x27;s just be square derided by a plus C Tim Scottson theater times that CO Cynthia. So let me rewrite that. So the point The lines England Petey just gonna be a square minus c square. Where did they see which is just the multiplication of distorted the addition of this too minus be square but by a plus. See temps go Cynthia times on extra co cynthia to remember that this term right here is just our And we determined that one right here. So now we have determined that we&#x27;re just gonna at all of these terms. I want to multiply our coast and fate over here. And remember that we said that B square is equal to a square of minus C square, so we&#x27;ll have b squared, divided by sea minus B squared co Cynthia, you better buy a policy Cosa stato. And so this is gonna be able to monitor right over here. So this term is just gonna be equal to be square. I&#x27;m just gonna multiply everything on the bottom. Sort of common denominator is just gonna be C attempts a plus. Seiko Cynthia Dempsey. So we&#x27;re gonna multiply everything r B terms is gonna multiply by this and this term is gonna multiply. Break this one. So be squares times a plus. See temps coast and theater minus c Tim&#x27;s be square cosign fado. So I&#x27;m gonna I&#x27;m gonna multiply everything through over here and I&#x27;m gonna distribute this. See? So we&#x27;ll have B square times, eh? Plus B square. Come see Tim Scottson theater minus C tends to be square times. Cosine theta he read it by 8 a.m. c plus C square, coast and freedom. And now we&#x27;re here. We can see the district terms cancel each other. So finally, I&#x27;m going to rewrite this. A speech. Square terms, eh? Nope. Sorry. Be square temps, eh? Minus. Oh, Sorry. Divided by eight m c plus c square cosa stato. So we have determined this over here. It&#x27;s a nice time that we take the ratio of the two of them. So we have the ratio of P F. Goodbye. P d must be equal to see the red light, Eh? So we&#x27;re gonna work with the term So be earth PF Sorry. She&#x27;s just one right here. It&#x27;s just gonna be able to be square. We&#x27;ll have b squared divided by eight plus seat temps Coast in a theater divided by be square temps, eh? Times see tempts a plus C plus go, Cynthia. So you can see right here that I just pulled one c term over here. One of her common sees And their reason to do that is because we&#x27;ll have B square, Tim, see times this term right here. A plus C plus co Santa divided by the square tem See times a plus See temps coast in theater So we can see that this to cancel each other and this just becomes one. So this term it&#x27;s just one. So they will have be square Tim seat divided by the square temp See, in these two A&#x27;s Sorry, this to be square are also one Oh, just any sorry. So finally we have demonstrated that the ratio of p f divided by P. D. Using our directories s X equals a squared. Where did I see he&#x27;s actually our directories Since our ratio of p f divided by P. D. Is equal to see the brighter by a So now on part B, we want to prove that the eccentricity so hee is equal to see the bite of a And so he is defined by the ratio p f divided by P. D. To the ratio of these two distances the ratio of the distance between the point off, any point in the lips to its foresight to the ratio of the point of the distance off the distance between the point and the directories. Hope sorry. And so in this case is ratio. We have proof over here that is sequel to see Divided by A with our directories. So this is equal to see divided by a So finally, when to prove that this line segment of the foresight to the it&#x27;s directories, which is represented by a LSO this is L It&#x27;s equal to a well by e minus. He divided by a said to do so we&#x27;re gonna take the line segment f l. So it&#x27;s false eye over here, which has co ordinate See, So remember that our folks I are located are plus or minus C coma zero. So in this case, it would take the postive. Suppose I she&#x27;s gonna get this one right here. You don&#x27;t have a just a coordinate of C. It is a distance See from the center sword directories is this equal to X is equal to a square. Even if I see. So with this information in mind, we&#x27;re gonna take a square, right? I see minus the distance, see from the origin. So distance right here is just a squared divided by sea. And so now we know that e the sequel to see Divided by a So in this case, We can also say that C e sequitur E times A and we can substitute that over here we&#x27;ll have a square divided by E temps, eh? Minus attempts A and so one of this ace cancel with the square. So we&#x27;ll have a united by e minus. E temps, eh? And in this case, we have proved that the line segment from the foresight to this directories is equal to a divided by E minus three times a. So this problem we have proved that the directories has co ordinates a squared divided by sea by proving that the line that the ratio of the line segments PG rated by P Sorry, pfc, But LAPD, it&#x27;s equal to see divided by a We also proved that e physical to see divided by eight by its definition which which is the ratio off the line segment between the point to folks. I focus in this case and point to direct tress which was just the result offer proof over here, which is just see divided by eight. By using this by using all Durant and the definition offer an ellipse. And finally we have proved that the lines that the distance between the focus and the direct tress is equal to a divided by E minus times, eh? By using the definition off its location in the X axis.

Charles Carter · Answer

Okay, so I&#x27;m gonna kind of do this. Ah, in a way that hopefully makes sense. Um, P uh, it&#x27;s just come to the square room of X minus C squared plus y squared and p d if we take the perpendicular distance to that line, um, which is gonna be a square? Never. See, They said we&#x27;re gonna have square root of X minus a squared. Oversee quantity squared. Okay, So if I set them equal toe the ratio of them equal to a constant because that&#x27;s how would go about finding that in geriatrics, where that constant is equal to see over a. But I&#x27;m just gonna prove that has to be able to see over a I can multiply both sides by P D and and square, which is gonna give me X minus C squared. Plus y square is equal to K square in case just to give me my placeholder times p d squared, Which is good thing you didn&#x27;t work it out, but X squared minus to a squared over see Times X plus a day of the fourth Ever see squared Using that my hyperbole equation toe that my wife squared has to equal X squared B squared over a squared minus piece where just something my hyperbole equation for y squared And I&#x27;ll get on my left side I&#x27;m gonna have X squared Money is two xy plus c Skrine plus x squared B squared over a squared minus b squared and that&#x27;s gonna be equal to And it make any changes there. Okay, so getting my X squared terms together Ah, if that&#x27;s attract everything on my right side if I move it all to my left side having equal to zero. Um, but in my ax I know is not zero. So I&#x27;m not like trying to quadratic formula anything. I just want to get all of my coefficients together, and then all my coefficients have to equal zero. I&#x27;m a show that by plugging and see over a and makes them zero Okay, So festive track. But case where that&#x27;s my only X squared term on the right. I&#x27;m gonna have X square. Uh, plus x squared times b squared over a square, Uh, minus K squared exclaimed as my X squared term. Okay. And then So when I plugged this in, I&#x27;ll have X squared times one, Cosby square number Ace burn minus K square. I should start looking like the East Interesting equation. So I have my case where it is equal to one plus b squared over a square to make it equal zero, uh, b squared over a squared and that can be written as a squared plus B squared over a square, which is he squared over a square. Uh, which, uh, for K that would be equal. See over a Okay. And I&#x27;ll leave the other two year, but they all come out to BC ever a Also doing the the ex term in the constant term. Okay, So Part B, it says, proved that the East interested e e is C ever a and that&#x27;s just I mean, that&#x27;s basically what I just did. Just that the Yeah. I mean, that&#x27;s that&#x27;s the definition of East. Interesting. And I just showed that. So I&#x27;m not too sure what the is asking for here. Um, just because the definition of the A centrist E is that that ratio would be constant for those points. And so I just proved that for all the points X y. So I can&#x27;t hang on to that uh, part C. This has proved that the distance from F to L is e a minus a every Okay, so the distance from after l it&#x27;s just gonna be see minus a squared oversee incidence. Eccentricity is equal to see over a, uh I mean, C is equal to a e. That&#x27;s that part. And this thing about it was just a overeat Scofidio a valid by e that&#x27;s really eagle to a over one times they oversee, which is a square never see, just gets that part or a Is that a overeat? There&#x27;s part c uh, for part B. I&#x27;m gonna look at it real quick. Yeah, I think again, like for part B. Um, we&#x27;re time of the East interest in East. Interesting is where the distance it&#x27;s the distance from. It&#x27;s our ratio. Basically what we just found the PF over P. D. That&#x27;s the definition of east. Interesting. And I just showed that that would be the constancy over a So maybe I did too much for partying, but, uh, my part B is is just included in part a. Okay, thank you very much.

Suman Saurav Thakur · Answer

we have a question we need to show that access choir by a Squire plus Y squared but being square equal to one. And the hyper bola. This is the lips and hyper bola access squired by a square minus y squared but be square. Well to one article. Electronic trees are total project remain means if a find slope of this divide by dx and slope of this let us right slope of tended on this which is uh if her lips and D Y by dx slope of hyper Ebola so productive they should be minus one. This is the meaning of all to the military secretary over here. So let us find dy dx differentiate with respect to X two x by a square plus two. Y. Do I buy DX Bible Square Equal to 0? Which means divided by D X will be equal to minus two X by a Squire into being squired by to Y. Two and two will get cancelled out. So do you have any excess hm minus X. B squired by esquire white. This is do you have any exit E. For ellipse? Similarly for help. Ebola. If we differentiate will be getting two X by a square -2 Whereby Be a Squire. Diva Body x equal to zero. Okay so uh D Y by D X will be equal to two x by a Squire into b squared by do it. These two will get cancelled out. So be esquire X by uh this is a Squire. Why this is D Y by the X slope of tangent at hyper bola. If we do like this we&#x27;d be getting uh minus X. Being squired by is quite why multiplied by be a square X by is quite a way. They should be equal to minus of one. Okay. Which means this is quite a way which means uh B squared B squared minus X squared equal to minus is square is square. Well y squared some mines mines will get cancelled out. So we have is it be as quiet right divided by a square is quite equal to Why? Square by Expert Equation # one. Now from any question let us get the value of access required by Y Squire. If he uh could just get the values. Mhm Okay, is everything good? Yes. Yeah. Let us get values from any one of these. So let us uh to get the values, we will be subtracting if this is a question number one, the same question number two. For the sake of simplicity will subtract any question number one from the question number two. The art form will be doing that technique. Question # two from the question # one. So this will be access choir by a square plus. We&#x27;re squired by being square equal to one and minus. Access square by square. Mine&#x27;s white square but it is quite equal to one minus plus and minus. So one will become right inside it will become zero and access square will be taken as common. So one by a square -1. But capital is squared bless. Why square will be taken as common? One by B squared minus one by capital B square equal to zero. Yeah. Uh so this is extra square and a square minuses square by square is what equal to Y squared one by b squared minus one by small, B squared Y squared more B squared minus capital B square being square. So we need to find the value of this. X squared, Y Y squared by X squared from here was squired by esquire will be able to is squared squared. Okay, we are dividing both sides by access. Quite so yeah, and this will be equal to be a squared B squared by a square square into is Squire minus a squared by small B squared minus capital base question Okay now we&#x27;ll be plugging in the value in equation number let us say question number three in question number three. So we&#x27;ll be getting could be a squared B squared by a square. A square equal to this being squared, B squared by Smalley square. Capital is squared square minus a squared by the square minus business. So these people get canceled out Here. It will be one. So the cross multiplying will be getting capital a square mind. Smalley square equal to a small base square minds Capital base square. Okay, so if we add capital D squared to both the sides, A squared plus b squared and again smaller square to both the sides. So this will become a squared minus B squared. So this is bro, thank you.

Regina Hays · Answer

Okay, let&#x27;s go ahead and try to step through this problem. Um, and were given that a, um we have a point. Pee on an ellipse. Um, and we&#x27;re gonna let, um, d be the distance and all, um, draw a picture here in Manila, draw a graph of the ellipse from the sinner of the Ellipse to the lying tangent. Um, at point p. And what we want to do is we want to prove that, um, the distance from point p to one of the folks I times the distance from point p to the second post I times this d square times the distance squared from the center of the lips to that line. Tangent at P, um is constant. Um, as P varies on the lips. Okay, Um and so what we&#x27;re gonna do is, um, we&#x27;re gonna get a little bit creative. Um, and so the first thing we do is we&#x27;re gonna actually defined the, um, basically lips as X squared over a squared plus, why squared over b squared and equal toe one, and some may go ahead and draw a picture, which we&#x27;re gonna have to keep coming back to, um and So here we go. Um, let&#x27;s just make this. I&#x27;m gonna try to make a bigger picture. Or so that&#x27;s negative. A in a, um, and this will be be and negative B. So here&#x27;s that the lips whips not drawn too great. And so in here is that point p. Um, and let&#x27;s go ahead and label that point p Let&#x27;s just label him x and Y, um, And then what we want is we know that we have, um, this line segment, um, that we&#x27;re gonna label d So the distance from the origin to that point and then what we can also do is, um if this is, um or these are the folks I And so we&#x27;re gonna let that be, um, first I want. And first I, too. And so if I actually draw a line segments to make things easy, I&#x27;m gonna label those as are too well, so this is gonna be or two. And this one is going to be Excuse me. Our what? So there we go. Um, And so what we want to do is, um, to say that, or one times or two times that distance squared is constant. no matter where I put that point. And so the first thing I want to do is to, um we know that one that, um four any lives or one plus, our two is equal to two way and remember bets, because it&#x27;s our one and our two or the distance from a point on the ellipse to each of the so size. So those the addition of those two distances will actually equal to a where is it up hyper below? The difference of those two distances equals to a okay and so and then, um, this is going to be the line tangent. So we&#x27;re actually gonna draw a tangent, whips I can get that drawn a tangent line, and we&#x27;re gonna extend him out. So here&#x27;s my tangent line. Okay, so let&#x27;s first on working on this or one times are two scenario, um, and see if we can kind of work that out. So we want to know what our one times are too is, and we&#x27;re going to actually be a little bit creative. Um, and I&#x27;m gonna say I want to know whether twice our one times are too is And, um, I&#x27;m gonna let that be our one plus or two squared minus R one squared, minus R T squared, Okay. And where I got that waas is because if I actually so I&#x27;m going to assign out here If I actually square this and I get or one squared plus, um, to, um or one are two, um, plus or two squared minus R one squared minus R two squared. You notice that, um, that this two or one this two are one are two, um, actually will equal, um Well, actually equal, um, this equation right here because when I do that, I actually get two, or one are two is equal to, um, or one squared andi, plus our two squared minus r one squared minus r t squared. And so, um, that will actually equal. So when I do that, these cancel so, uh, to to our wine are too. Um and I don&#x27;t want to do this, so that will. Actually, these two will add up to zero. These two will add up to zero. And so this side actually does equal to or one or two just kind of in a different way. Um and so then that means what we have here is that or one times are too is 1/2 that. Well, we also know that or one plus our two is to a and so this actually becomes 1/2 times for a square minus R one squared minus R two squared, OK, and so now I&#x27;m going to kind of look at what? Or one and our two actually equal. So that means I&#x27;m gonna have to go back to when we have a blank page, actually go back to, um, our picture. Um, right here. And I notice that, um, are one if I draw a right triangle, here are one is the high pot news for this triangle right here, which is haIf is why? And it&#x27;s based length is actually, um, um, going to be wth e x value plus See cause. Remember, F one is negative, C comma zero and F to iss C comma zero. So this distance is X plus c. So this is a high pot news. And so, um, are one squared are one squared is actually the length of that high pot news. And so that is going to be minus X plus c squared, Um, minus y squared. And so now I&#x27;m gonna go ahead and look at the second triangle, which is right here, and I notice that once again, this is a right triangle. Where are true? That high pot news of the triangle that is created from why And this distance right here and that distance is actually going to be X minus C cause it&#x27;s gonna be this, ext I hear minus C. And so now what I have is this is minus X minus C squared, um, minus the y squared. And so this is this actually is, um, or two squared And this is our one squared. Um, And so now I&#x27;m going to go ahead and four things out on combined light terms and so forth. So I get 1/2 for a squared. Um, this is minus X squared minus two XY minus C squared, Um, minus y squared minus X squared plus two x c minus C squared, minus y squared. And so you noticed that the two x Ys add up to zero and then I have on this is equal to 1/2 times for a squared. Um, minus two x squared minus two C squared, minus two y squared. And so this is actually becoming to a squared, Um, minus C squared, minus x squared, minus y square. Okay. And now we do know that C squared is a squared minus b squared. So I&#x27;m gonna pull in that factor as well. And so, um, are one times are too is actually coming out to be, um, a squared plus B squared, minus x squared, minus y squared. Okay, so we&#x27;ve kind of gotten that so far. Um, And now what we want to do is to look at, um, go back and look at, um, this d and this tangent line. And so what I&#x27;m gonna do is to help us. I&#x27;m gonna go go ahead and read all that picture. So this is critical. We need to keep it on that. And so I&#x27;m gonna go ahead and read all that picture and try to clean it up a little bit. Um, And so now we&#x27;re actually gonna focus on if this is the Elliotts and this is that point p at X one, Will x and Y, um, And then if this is that tangent line and this is D. Okay. You notice what is happening is I&#x27;m now have kind of like this right triangle right here where this the tangent line is becoming the iPod news of this right triangle right here. And D is the distance from the origin to the high pot New. So that&#x27;s actually going to be the height of that right triangle. And so I&#x27;m gonna utilize that to kind of help me out. So the first thing I need to do is I&#x27;m talking about a tangent line, so we know we always have to take the derivative. And so the derivative of why, with respect to X of the original equation X squared over a squared plus y squared of a B squared equal toe, one is going to be negative. B squared X over a squared. Why? And this is at point X. Why? Okay, um now, um, because this is a slow and I noticed I want to get these values right here. Where this tangent line intersex, Uh, the Y axis and eggs axis. And, um, just intuitively, because of because of this slope are here. We know that this point right here is going to be a squared over X comma zero. And this point right here is going to be zero comma B squared over. Why? Okay, so we have those two points. Now, what we&#x27;re gonna do is I still need to try to develop a relationship with this, um, devalue. And so this is a right triangle. And so if I actually look at that area of that right triangle, it&#x27;s gonna be 1/2 the base length, which is a squared over X squared times a high, which is b squared over. Why? Okay, but we also know we also know that, um, another way to find the area of this triangle right here is this is the high, and this is the base length, and so those two actually have to equal each other. So this will actually equal 1/2 um, the height, times the base length, which is that length of that high pot news. And so that is going to be a squared over X squared. Plus, um, b squared over. Why Squared. Okay. And so now I&#x27;m just gonna kind of utilize these two things to sell for D. And so when I do, um I get, um, one I get, um, d is equal to that. A squared of times b squared over the x. Why? And then, um, over that square root of a to the fourth X squared plus beat of the fourth, Why squared and then my square both sides because we actually want that d squared. And so this becomes eight of the fourth beat of the fourth over. Um, and then a man have, um, eight of the fourth, um, eight of the fourth. Why squared plus B to the fourth x squared. And what I did was when I squared it, I get X squared y squared out here, and then I distributed Okay, um and so now what I need to do is kind of be creative about certain other things. And so what we do know is that, um, that point x y lies on the lips. Um, And so, um, and the Ellipse equation is x squared over a squared plus y squared of a B squared equal to one. Which another way to write that is that b squared, X squared. Plus, um, a squared. Why? Squared is equal to a squared B squared And so what I can do is one we can multiply both sides by, um, multiply both size by, um, a squared. And when I do that, we get a to the fore Why Squared is equal to eight of the fourth B squared minus a squared B squared x squared. And then the second thing is, we can multiply Go back to the original and multiply by B squared. And when I do that, we get, um, beat of the fourth X squared is equal to, um a squared beat of the fourth minus a squared B squared y squared And so we&#x27;re gonna use thes to substitute in here. And so what happens with that is I get D squared is equal to eight of the fourth beat of the fourth over. On we get, um, we haven&#x27;t eight of the four b squared minus a squared B squared X squared, plus a squared beat of the fourth minus a squared B squared. Why squared? Yeah. And so that&#x27;s when I substituted those two values in, and you notice that all of them now have some kind of form of a squared B squared. So I&#x27;m gonna go ahead and factor out on a squared B squared. And when I do that, I get, um, a squared minus X squared plus B squared, minus y squared. And so this reduces down to a squared B squared over X squared plus B squared minus a squared plus B squared minus X squared, minus y squared. Okay, which is looking kind of familiar, because now, um, or one times are too times d squared and are one is the point. The point p from, um, focus one. And our two is the point p from focus, too. We what? We remember that hopes that, um where was that? That are one was somewhere. Um oh, was right. Here is a squared plus B squared minus X squared, minus y squared. And so this is gonna be, um, a squared plus B squared minus X squared, minus y squared. Because that&#x27;s what or one times are too waas times. Now that d squared, which is a squared B squared over a squared plus B squared, minus X squared, minus y squared. Uh, so you knows that this just simply reduces down to a squared times B squared, which is going to be constant for any lips. And there we&#x27;ve, um then the problem

Joshua Eastwood · Answer

that were, as thio find are proved that the equation of an ellipse with center 00 and no guy at zero and plus or minus C is gonna be where a is greater Zordon see which is greater or equal zero is gonna be this. This is what we&#x27;re trying to prove that it ends up being why squared over a squared plus X squared or B squared is equal one. That&#x27;s what we&#x27;re trying to prove and we&#x27;re trying to prove it, given that the other piece of information gives us over. Here is the B squared is a squared minus C squared. So we&#x27;re gonna look at it in this direction. Then you have a center, you have a focus here, and you have a focus down here. And then you have a vertex at the end out here and a vertex at the end out here, okay? And so what we&#x27;re gonna notice is that the definition of in the lips is this is the scent of all points such that this is true. And so the distance here in the distance here is gonna be defined by the distance formula, which is the square root of X minus Something squared in this case zero because these air all on the zero you notice that x zero there in X zero there. So the distance from here to here is going to be X minus zero squared and then plus why minus c squared And then we&#x27;re gonna add the same distance down here, except it becomes X minus zero squared. Plus why minus minus c. So why plus c squared is equal to a All right, that&#x27;s what we&#x27;re gonna start. And then what we&#x27;re gonna do, I&#x27;m gonna just jump to the next page. He is. I&#x27;m gonna write that first value, that out of the square root. And I&#x27;m just gonna leave it alone with the square root of X squared. Plus why minus c squared and I&#x27;m gonna bring the other one over here. So, to a minus the square root of X squared plus wide plus C squared, right? All I did was just bring this term over there, right? And then I&#x27;m gonna square both sides. It&#x27;s when I do that I get X squared plus y minus C squared. And over here I square this term for a squared I square this term. So I get plus X squared plus y plus c squared. And then I have an extra middle term, which is two times that times. That&#x27;s what ends up being foray under X squared plus wide plus C squared. Then I am going to expand, so I have X squared. Plus, this expands its y squared minus to see why plus C squared and me a four, a squared plus X squared plus. And then this expands is why squared, Plus to see why Class C squared and then it&#x27;s minus for a square root of X squared plus y plus c squared a lot of terms here, but what&#x27;s nice about this is you notice there&#x27;s an X squared here and there&#x27;s next. Where there there&#x27;s a Y squared on each side of the equation and there&#x27;s a C square on each side of the equation. So all of those terms disappear. And that leaves me with just this guy, this guy, this guy and this guy just four terms. I want this turn to be positive this term here, so I&#x27;m gonna move it over to the other side equal sign, and I&#x27;m gonna bring this one back. So what that looks like is or a square root of X squared plus y plus c squared is equal to And I have four a squared and then plus to see why and then plus another to see why. All right, so that means I get this for a square root of X squared plus y plus c squared is equal to. And then you see why on Tuesday white ad together. So I get for a squared plus four. See why? And then what I&#x27;m gonna do is I&#x27;m gonna drop the fours because there&#x27;s one on each term and then I&#x27;m gonna square both sides. So when I do that, I get a squared multiplied by X squared, plus y plus c squared. And here I get eight of the fourth, plus, uh, to a squared. See why plus c squared y squared. Okay, you&#x27;re gonna get us a perfect where now that gives me a squared front X squared. Plus why squared? Plus, to see why plus c squared. It is equal to eight of the fourth plus to a square and see y plus c squared y squared. So then I&#x27;m gonna yet Eastern. That&#x27;s weird. Plus aged, buried white Plus who a squared you Why? Plus a squares. Each bird is equal to 1/4. You a square it see why C squared y okay, long terms there overly. You&#x27;re following. I know it&#x27;s a lot of Eldora, but what we&#x27;re noticing, then hopefully you&#x27;re noticing with me is that there&#x27;s a Y squared term here. There&#x27;s a Y square term there. So I want to get those together. And then these two terms here are like, terms. I want to get them together. And if I can get those together, then there&#x27;s also this guy, all right? And this guy that have a squared in them so we can figure all these together to get this a squared minus C squared times. Why squared? Okay, that&#x27;s bringing these two terms together right here. This guy and this guy spring them over, and that&#x27;s gonna be a plus. And then there&#x27;s my a squared, X squared term. Hey, and that&#x27;s gonna equal. And then I&#x27;m gonna bring these two guys over together. No, a squared onto a squared minus, C squared, right. And what&#x27;s gonna happen? Actually, I should notice these. I didn&#x27;t actually cancer the mood, but they will because they&#x27;re on each side of the equal sign. So this a square here and this ain&#x27;t the fourth are coming together. You give me this because my multiply this I got aced the fourth minus eight grade C square Right now that a squared minus C squared is equal to B squared. So I&#x27;m gonna replace it. So I get b squared. Why squared lost a squared X squared is equal to a squared b squared. And then finally, what I&#x27;m gonna do is that we divide everything by a squared b squared. And when I do that, I&#x27;m left with y squared over a squared plus X squared over B squared is equal to one, and that proves the equation of the Ellipse. So it&#x27;s a long process, but hopefully, if you go through it and listen to its own applies at times, you understand where it&#x27;s all coming from.

James Schroeder · Answer

Well, they&#x27;re in this problem. We&#x27;re esta derived. The equation, friend lips. It&#x27;s a challenge problem. So it&#x27;s a little bit time consuming, but we&#x27;re gonna start with what we know about a general eclipse, and then we&#x27;re going to see how to get to the standard form equation for an ellipse. So we&#x27;ve got a an ellipse with the folks I at si com a zero and minus C comma zero. The definition of an ellipse says that any point on that ellipse the distance from one focus and the distance from the other Focus added together is a constant value. And then the problem they said they wanted to set that constant to to a in A if you&#x27;ll remember, is also half the length of the major axis. That would be a distance there. So what a gun is a triangle drawn from the folks I to a point on the Ellipse and label those D one and D two. So what we&#x27;re being, what we&#x27;re saying is we want d one plus B two to be constant to a and that&#x27;s just the definition of an ellipse. And so, if I know that, then I confined what d one and d two are using the the Pythagorean theorem. Remember that a squared plus B squared equals C squared is a Pythagorean theorems. We&#x27;re gonna use that to fill in D one and d two. So first D one is gonna be, um, square road of X minus C that&#x27;s X minus and negative. See, So it&#x27;s gonna be X plus C squared plus y squared. We&#x27;re gonna add D to D two is gonna be X minus C squared plus y squared. That&#x27;s gonna equal to a That&#x27;s our basic equation that just says that the distance D one and D two added together equals two way. And from here now we&#x27;ve got it set up. We&#x27;re just gonna do some, uh, do a little bit Algebra two. I saw this. So let&#x27;s take this. Take this term and move it over to the other side. That&#x27;s gonna give us squaring of X minus C squared plus y square equals two a minus square it of x plus C squared plus y squared. OK, And now let&#x27;s take that and we&#x27;ll square both sides and we&#x27;ll end up with X minus C squared. X minus C squared plus y squared on this side is equal to remember We&#x27;re gonna square this entire expression and it&#x27;s got that negative sign in there. So we&#x27;re gonna have to square the entire expression we&#x27;re gonna end up with is we&#x27;re a squared minus or a square roots of X plus c squared plus y squared plus x plus C squared, That&#x27;s why Square So now we can we have some common terms on both sides so we can eliminate Why squared from both sides? Well, then we can expand our expressions of one more time. We can take this term and squared this term and square it So we have X squared minus two XY. Let&#x27;s see squared equals for a squared minus four A Where, uh, of x waas See squared plus y squared plus X squared minus two xy los C squared. Now again, we got some like terms on both sides. We have an X square that we can cancel out. We have a C square that we can cancel out, and then we can rewrite that expression. So you keep your notes, I&#x27;m gonna race this stuff on the top to be able to do this in a faster way. Okay, then we&#x27;ll move that expression up to the top, and we&#x27;ll keep working. It&#x27;s really just algebra. Just keeping track of what we&#x27;re doing and keeping everything organized. Coming back up to the top one of minus two x d. Remaining on that side equals for a squared minus four A. I&#x27;m square it of X plus C squared. Plus why squared the minus two x c? So that&#x27;s re written. I&#x27;m gonna get rid of that on the bottom two. Yes, Um, space to work. Okay, now I can go back. Teoh. Next time, let&#x27;s rearrange our terms. Will put the move this term over the left hand side of the equation and will combine, uh, term on a here and year. Bring that over. So what does that leave us with? That leaves us with for a on this side. I&#x27;m just square of X plus c squared plus y squared. That&#x27;s equal to four A squared. Uh, that should have been a plus two. I see. So plus two xy buster is plus for X c. Okay, so now again, we want to get rid of this square run and surround the square, both sides. So let&#x27;s take both sides and square him again. And what we end up with we end up with is well, before we do that before we square, we can also eliminate the all the force we divide all terms by four. You&#x27;re rid of that. They may serve math a little bit easier. Makes arithmetical leisure. So now we&#x27;ll square these terms. We end up with a square times a quantity X plus C squared plus y squared equals eight of the fourth us to a squared xsi waas x squared C squared. We&#x27;ll expand out our X plus c squared on this side and rewrite it all. Get a squared X squared waas Teoh a squared C X plus a squared C squared, plus a squared. Why square? That&#x27;s just distributing that a squared through all the terms. Clean that up. That&#x27;s equal to eight of the fourth, you know, plus to a squared X c plus X squared C square. Okay, thankfully, we need Teoh eliminate a couple of turns this to a squared X c to a squared C X. That&#x27;s the same things We can eliminate that And can we write our equation. We can move things around. Rewrite it. So we&#x27;ve got a squared X squared minus X squared C squared. Let&#x27;s say squared Y squared is equal to eight of the fourth, minus a sward C squared. Okay, again, I&#x27;m gonna eliminate clean up a little bits weekend rewrite our equation. Yeah. Okay. It&#x27;s now rewrite at a squared X word minus X squared C squared, plus B squared. Why squared what equals eight of the fourth minus a squared C squared. Okay. You know, I&#x27;m gonna get rid of that on the bottom. We rewrote it there. Okay, so now we can take the next we factor out any squared from the right side that&#x27;s gonna equal a square times a a squared minus C squared for your factor out in a squirt from the left side. Also, we get a squared times x squared way Backtrack a little bit, manufacture out the X squared here. Factor out the X squared term here X squared im se squared minus c square plus a squared y squared. Okay, Now we want to divide each side all the terms by a squared times a squared minus C squared. You know as long as we do that to all sides, then what&#x27;s We&#x27;re staying legal now on the right side, top and bottom Both cancel my left with one on the left side, a squared minus C squared cancels from the first term we end up with X squared over a squared in the second term are a squared cancels I left with Why squared on top? Why squared over a squared minus C squared? So that&#x27;s the first part of the problem. One is to show that the equation is X squared over a squared plus y squared over a squared minus C squared equal to one. Then they told us in the problem toe let b squared equal a squared minus c squared. So let&#x27;s just do some substitution. Now we can take this quantity is the same as this quantity so we can substitute there. Move this over So we have a little bit of room to work So now we can rewrite. With that substituted, we can rewrite this as X squared over a squared plus why squared over b squared equals one that is a standard form for our lives. So it&#x27;s really just

Problem

Consider the hyperbola $$\frac{x^{2}}{a^{2}}-\fr…

Problem 58 Hard Difficulty

Answer

a. $\frac{P F}{P D}=\frac{c}{a}$
$L$ is a directrix
b. $\frac{c}{a}=e$
c. $\frac{a}{e}-a e$

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Franklin D. Demana,Bert K. Waits,Gregory D. Foley,Daniel Kennedy

Precalculus: Graphical, Numerical, Algebraic

Grace H.

Heather Z.

Alayna H.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

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a. $\frac{P F}{P D}=\frac{c}{a}$$L$ is a directrixb. $\frac{c}{a}=e$c. $\frac{a}{e}-a e$

Precalculus: Graphical, Numerical, Algebraic

Grace H.

Heather Z.

Alayna H.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

Franklin D. Demana,Bert K. Waits,Gregory D. Foley,Daniel KennedyPrecalculus: Graphical, Numerical, Algebraic

Grace H.

Heather Z.

Alayna H.

Kristen K.

a. $\frac{P F}{P D}=\frac{c}{a}$
$L$ is a directrix
b. $\frac{c}{a}=e$
c. $\frac{a}{e}-a e$

Franklin D. Demana,Bert K. Waits,Gregory D. Foley,Daniel Kennedy

Precalculus: Graphical, Numerical, Algebraic