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Find the focus for the given ellipse x2 + y2 = 9(x,Y)...

Question

Find the focus for the given ellipse x2 + y2 = 9(x,Y)

Find the focus for the given ellipse x2 + y2 = 9 (x,Y)



Answers

Find all points on the ellipse $9 x^{2}+25 y^{2}=225$ that are twice as far from one focus as they are from the other focus.

And this problem will give the equation of the old lips and we'll be looking for all the important pieces. So we have nine X squared plus four y squared plus 90 x minus 16. Why? Plus 2 16 equals zero. So the first thing I'm going to do is I'm gonna move this to 16 over the other side, making it negative. The second thing I'm going to do is I'm going to focus on the X is I'm gonna factor out the leading coefficient, which gives me nine times X squared plus 10 X with a little bit of space that I'm gonna focus on the wise benefactor out the leading coefficient, which is four. Which leaves you with y squared minus four y I can now complete the square. And where to write this equation in standard form to find all the pieces five square five when it cut 10.5 again. 55 squared is 25. But I'm not gonna add 25 to the other side. I'm going to multiply nine times 25 add to 25 to both sides over here. I'm going to take negative to cut it. Negative for Cut it in half to get negative to negative. Two Squared is 44 times four is 16. So in this case I have nine times X plus five quietly square plus four times why minus two Quiet scored when I factor negative to 16 plus 2. 25 plus 16 is 25 which will leave me with the equation X plus five squared and then I have 25 9th. Why minus chew squared and I have 25 4th is equal to one. This is equivalent to X plus five squared over 2.78 Why minus chew squared over 6.25 is equal to one. So what does that tell us? Well, first of all, tells us the center of my lips. The center of the ellipse comes from the numerator. It's the opposite values. So negative five to so it negative 52 is the center of my is the center of my lips. 6.25 is larger than 2.78 So if I take the square root of 6.25 we're gonna get 2.5, which means I'm gonna go up 2.5 and down to an ad to get to my vergis ease, which will be at negative five, 4.5 and negative five negative one half to find the left on the right, I'm gonna do 2.78 square to that is gonna be 1.6 ish, 1.7 ish. So it's gonna be about here and about here. They don't ask me for the, uh, coordinates. I'm not gonna give him to you. The only thing we need to find now are the folks I. So the folks out here, if I take 6.25 and subtract 2.78 and take it square root, I will be going plus or minus 1.86 up and down because the longer access is up and down. So in this case, it's negative five, and then I'm gonna started to and I'm going to add 1.86 So I'm gonna get 3.86 and I'm gonna have negative five, and I'm gonna start at two. I'm going to subtract 1.86 which is gonna be a 0.14 So about here about their Here's my picture with my folks, I the coordinates of my folks. I'm Ivor Theses and myself

Parabola which is giving us X squared plus y equals 200. We can write this expression as extra square equals two minus off. Why minus under which is Xer Square? Because toe negative off four times one divided by four times. Why minus Under What information we're getting here. This is aboard. So this downward this is excess square is this it is downward parabola don would open parabola with the vortex we have Zittel, I wondered and focus which is Zittel 100 minus one divided way full So 100 minus one, divided by four which is zero 399 divided Well before If you draw this parabola xx iss why excess The vortex is zero comma 100. So let's say this is a vortex. 0100 So parable lies in this direction. The focus is here. Zero 399 Do you herd way for In the given caution, it says we have to find the lips, its shares the vortex and focused image of this parable. Also four lips We have one word exists with one focuses this one and second focuses. Or is it? This is a second focus which is 004 day lips, and we have to find the food that is second vortex. So let's safe will draw the it lifts, passing through these points so focused one is little. 3994 and focuses Little Joe So center of this ellipse is made point off. Let's you focus. It is F one, and the second floor was It is have to so midpoint off F one and F two, which is zero plus zero. Divide by two Koba zero plus 399 divided by four Only one way to which gives the value was little 399 divided by eight. So we have This is a Sando off this that lifts center, see, is little 399 Divide. Wait Now alerted the next word. Excited on Why access so X coordinated with the same. And Vikrant, I don't know. So we should find Let's say it is white. Later zoom. So now the center is always the midpoint of the vortex. So let's say this vortex A and B. So we know the Sando is midpoint off The vortex made a point off the line, joining the vortex what we have. So we can write center is zero 399 divided by eight It closed toe zero plus zero Divide by two comma. One is 100 another war Texas way So Y plus 100 divided way to this gives us why plus 100 divided by two is a course toe +399 The world way eight It is two times for from here Why Valley We are getting 399 divided by four miners 100 which is minus one divided Grateful So this second vortex we have negative zero comma niggardly one divide way For now we have to find the equation off the ellipse. What we can write a value we know from the center This distance is a so value off he is under minus off 399 divided by eight So 800 minus 399 That is four little one you've heard by eight the value of B We have to get a total value of this distance. I see you've given us 399 divided by four minus 39 90 wide weight that is 399 divided by four minus 399 Divide way eight which is 399 divided by eight correspondingly. Well, get the value of B. The square root off C square that is a square minus C square, which is, and let's they will try to find the value of B Square only, but just four little one square minus 39 90 square, which is it? Closed toe 1600 divided by 64 and he's quit is 4012 square is 16 little a digital one. You know everybody 64. The question of this ellipse. We can like excess square you heard by four Little one divided by eight hole. It's Quest les Y squared. Divided by it is 16 100 so and this is a vertical lips, so it should be on the no matter off white. So 401 divide by eight bullets where it is 1600. So 40 divided by eight hole is quest equals toe. This is required legs at the vertical so denominator off Why should be greater And this is honestly

For this problem. We need to find the equation for an ellipse. Given some basic information about the Ellipse, we know the eccentricity, you know, one of the direct tresses and the corresponding focus. So putting all of that together, we need to find the equation of the Ellipse. Well, let's begin by graphing what we know and see what we need to find out in order to write our equation. First, I have a point. So we could put that point in. I have a focus at the 0.0.0.40 So that's right there on the X axis. No, What? We don't know. We need to be careful not to make the assumption We're not told where the center of this ellipses It might be at the origin, but we can't make that assumption. Okay, so I'm just gonna kind of put that out there, But we do know where the focus is, and I know where the director exes. It's here at X equals nine. What's right there? Okay, because the Director X is X equals and we see where the focus is, and I have no idea if that eccentricities right. I haven't looked at that yet. I'm just doing a generic picture of the Ellipse here. It is going to have the X axis as the major axis. And we know that because of where the direct tricks is, the directors is X equals. So the major axis is also going to be the X axis. Okay, so that tells us what the format of our final answer is gonna be. Just knowing which access is the major axis. Okay, because remember, for an ellipse, we're gonna have X squared and y squared is our numerator adding them equaling one. But where the A and B goes, determine, you know, is determined by the major axis. Since our X axis is the major axis, that denominator will be a squared. Why squared is B squared and a squared is gonna be the biggest one here. Okay, so that's our general format. If I know A and B, I can plug them in and I'll have my answer. But I don't know either of those yet, so we're gonna kind of leave that there. We know that starts. Our goal is to find A and B Okay. Before we leave the focus, let's see what that tells us because focus goes along with C. Now, remember I said, I don't know where the center is, so I'm going to put a little X here. The X may not be in the right spot. It could be right on the origin. It could be on either side. I just need a place for it. And I'm going to say that this I'm gonna use em. It's just because I've got lots of A B's and C's floating around and I don't want X's and y's because I got those floating around. So I'm going to say that our center is at M zero. I know the Y is zero because the X axis is the major axis. The focus is right on the major access there. Okay, so that's my point. So C is going to be the distance between four and M. I talked about the distance of the difference between them. That is subtraction, so C equals four minus M. If M is zero, if it's at the origin, then see is going to be four. If it's elsewhere, that's going to adjust my C. Okay, so that's a much as I could tell from the focus right now. Let's take a look at the eccentricity next. Okay? Eccentricity is a measure of the roundness of the ellipse, and we measure it by taking the ratio off. See, over a now, remember, A is always the biggest for an ellipse in the A, B and C So put the biggest number of the denominator things fractions always somewhere between, um, zero and one. In this case, we know that it equals two thirds. Now, that doesn't mean that C equals two and equals three. Just means that's our ratio. Okay, so I can say that if I cross multiply three c equals to a So if I need to write one in terms of the other, I can. So that's what the eccentricity tells us. What about the direct tricks? Well, the direct tricks is going to be it's going to be mm plus or minus a over E. Now, remember, I have to have that em in there because we don't know for sure that the center is actually at the origin will be very nice if it was. But we don't know that for sure now, because I know, uh, my director X is nine. The other one will be over here at negative something. I know that I could take the plus here because I'm gonna take taking my center plus overeat to get to nine, not minus. So I'm gonna erase this plus or minus and just put in a plus. Because in the context of what we're doing, the directors we have is the right hand one. It's going to be a plus. Okay? And I know that the director X equals nine so I can plug in a nine there a over e. Well, the eccentricity is two thirds. So if I divide by two thirds is the same as multiplying by three halfs. So I have nine equals M plus three a divided by two. Now, the problem is I'm going to raise this other directorate, so I got a little more room. Problem is, I've got two variables here M and a So I want to rewrite one of these in terms of the other. Okay. And I'm going to actually use our piece of information here that c equals four minus m. Because I could put see, in terms of a I have and I'm gonna put a red circle around it. So it stands out. I know that three C equals two A. Which means if I solve that for C. C equals two a over three. So I could do that substitution there so I can solve this. I now have two equations in terms of M and A So this one that I just wrote. If I solve this for M, I have m equals four minus two a over three, and I can plug that into the equation that I just box at the bottom. I could say I got nine equals m, which is four minus two way over three plus three a over to let's combine our like terms get a common denominator. So pull that four over it equals five. My common denominator will be six. So I've got negative foray over six plus 9 8/6, which is five equals five a over six solving that gives me a equals six. Okay, so if a ISS six, then if I look and see what see is I can see that see has to equal for because I know how they relate to each other with a a six, then by this red circled equation. So you would have to equal four. Well, but C equals four minus M. Which means M is zero. This is fantastic. That means I am centered at the origin. Okay, If she wasn't equal to four, it means I'd be offset a little bit. So I know we did a little extra work by assuming that we weren't at the center. But that way we could be sure of our answer. This is definitely centered at at the origin. Okay, so I'm going to scroll down just a little bit. Just give us a little tiny bit more room. Since I know A and C, we confined Be, don't forget. Don't lose sight of the fact that what we're trying to find no way A and B will let us write our equation. So a is always the biggest number when it comes to an ellipse. So I get a squared equals B squared, plus C squared. A squared is 36 c squared will be 16, which means B squared is 20. So that's all I need to know to write my final answer. So I've got X squared over a squared, which is 36 plus y squared over B squared, which is 20 equals one. So this is the standard form for the equation. Uh, that has all of the, uh, information that we were given. The eccentricity, the directorates and the focus. This is the equation we're looking for.

Mhm. So for this problem we're trying to find the coordinates of the direct tricks and the focus. So let's start off with what we're given, which is three X squared minus nine Y. Of course. Yeah. Now this does not look like standard form and it isn't and it's best for us to get it into standard form. It may not seem important, but I assure you it is because it will allow us to really easily find these important parameters. P. So first thing we're gonna do is we're going to add nine Y to both sides, and we can divide three on both sides. So now we have officially gotten into standard form. Then what we can do is, yeah, we can view this as the same thing as four times 3/4 house wine. That tells us that the focus is going to be 0, 3/4 and the direct tricks is going to be X equals Y equals a negative 3/4.


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