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Sketch the graph of the equation. Give the coordinates of the vertices and foci.$$4 x^{2}+9 y^{2}=1$$...

Question

Sketch the graph of the equation. Give the coordinates of the vertices and foci.$$4 x^{2}+9 y^{2}=1$$

Sketch the graph of the equation. Give the coordinates of the vertices and foci. $$4 x^{2}+9 y^{2}=1$$



Answers

Find the vertices and foci of the ellipse and sketch its graph.

$ x^2 + 9y^2 = 9 $

Okay. This problem gave us an equation. Wine minus four square over 16 minus thanks. Plus two squared over nine. I was buying it wants us Thio. Find all the special characteristics of this hyperbole. And then to graph it for the special characteristics we need, we need thio find what a B and C are became. So A and B are just the square roots of minds. The dominators. Okay, so my value is just gonna be It's word of 16 which is for be value be the square root of nine, which is three in the sea. Value will be the sum of the squares and then take the square root. So 16 plus nine is 25 the square root of 25 is fine. Okay, Um, next thing to notice is that this is gonna be a vertical hyperbole in the center. We'll be at the point. Negative two for okay. So because the center is that negative to four, and it's a vertical hyperbole, and my vertex will line up right on that negative too. So the both bird Texas will have the X coordinate negative too. And then the white corn, It will be a units away from the center. So a units four units away from four would be eight and zero. Remember, Texas, my folk I and are going to be seein it's away from my center S O. They will also be on negative, too, because a vertical hyperbole. So the first focus will be a negative to nine. A second focus will be a night to negative one and then my ass in tubes First off the slope, it's gonna be a plus. A moment to end. It's gonna be a plus or minus. Ah, a over B for vertical So before over three. So the equation of my ass and tits are both gonna be why minus four equals cluster minus 4/3 exposed to you know, the graph it. I'm just gonna go to the second son. So my center, there's gonna be a negative to four, get myself a little bit of room. So because the center is a positive value, give myself a little bit more room on the positive sign and my X ray will be negative. Someone get myself just a slightly a little bit around towards the 95 side. So the market. My center with an axe, Centurion. Negative to four. Said that will be right here. Well, go ahead and draw my ass into so my ass and tits have a slope of 4/3. So we're gonna go up for over a serene went down for over three and same thing going the other way. Yes, There's my ass intense. I'm Ivor Texas are gonna be at eight. As native to eight and native to zero. Focuses will be in. No, you're too negative. One and number 29 So my heart purple. Uh, we'll leave from my bar, Texas, and approached assassin tunes. Thank you very much.

Who gave us the equation? X plus one squared over four minus. Why? Plus three square It's over nine is equal to one. This hyperbole is already in standard for him. We know that it's gonna be a horizontal hyperbole and the center is gonna be a negative one. Negative three. Our next objective is to find our A B and C values. Hey, and beer Just the square roots of our denominators s O a. Is this a square to four witches, too? To be is a spirit of nine, which is three. And so is the square root of the some of the denominators. So for nine is 13 so see is just rude. 13. This is a horizontal. Um hi problem. So that means our y values won't change. S o r ver. Texas will be of the form negative one closer minus two, which is our values in the negative three. So they run plus two. It's just gonna be at one of the three. A native on minus two B negative 3 93 These aren't even Texas for food guy. We'll follow the same format, started negative one, and we're gonna add and subtract for two points there, screwed of 13 and they will lie along the line. Y equals three. Number three. Okay. And then our ascent oats bray, horizontal hyperbole, the Aston Totes or be over a the slope is and the center I'm gonna use H and K because I already know that it goes through that point. That sentence well certain have wine. Plus three is equal to closer minus our scope is be over. And so it's just 3/2 Times Express line in the sea. A question of our ass in tips. Okay, so let's let's get a graph of this down on the next line because the center is not at the origin. I'm just gonna put an X with center. It's that negative one negative three. Our first task is to draw the Burgess sees the courtesies heir to a Wayne on a horizontal axis. So our first tour to see will be here in our 2nd 1 Here, um, are focuses are gonna be rude. 13 away from negative one, which is closer to about a little bit less than four, cause skirted for 16 a little bit less than four away from the center point, and then our Assam toads have a slope of cup 3/2. Um says, look, something like this. Excuse my abilities for making straight lines. Okay? And then started your courtesies and destroyer hyperbole approaching those estimates we just drew. Okay, thank you very much.

So we have ah X squared over nine plus y squared over four is one, and we're being asked to find the center. So the center here is 00 courtesies. See here. So a squared is nine. So x looks like the major axis. So in that case, Vergis ease our 30 and minus 30 since a is three now, the focus I foresee, uh, so C squared is a squared minus B squared. Remember, C squared is a squared minus B squared and B is four. B squared is four. So that's nine minus four. So C is route five. So in that case, through 50 and minus root 50 Now, remember, B is too. Since B Squared is for now, let's start graphic. So x is the major axis. So our vergis ease our ah, minus three here, and three and the Alps will cut the y axis here at, um, zero to zero minus two. That's minus 30 30 And we have the folksy will say here, uh, which is minus root 50 and route 50 So now let's draw the ellipse here roughly an ellipse, but not really, but make sure you label those important points

So we're being asked to analyse each equation. So we have X squared over nine plus two y squared over four. So since the number under X is large, that means we have a major access along the X axis. And since there's no ages or K's being added, two extra why we know that this sector 200 So let's get that out of We have a center at zero. Um, uh was zero. We also have. So now we know that the equation is going to be X squared over a squared plus why 20 over Big squared is equal to one. Since this, uh, let's lies along the x axis. So a square is going to be equal the nine, Which means that a we just the distance from the center to the Vernet for taxes going people three. So that means the Burgess sees are going to be a plus or minus three comma zero are, um, folk. I to find her focus we need to find would be is so b squared is equal to four, which means that B is equal to to. So C is going to be equal to the square root of a squared minus B squared. So C is going to be equal to nine. Minus four says could be equal to the square root of five. So that means that our folk I are located at plus or minus the square to five calm a zero. So let's draw this. And now that we have that we can grow. So we have our center here we have our Vertex Negative three and three and be the covert sees we're gonna be located at plus reminds too to negative too. And her folk I are going to be in between. So the square of nine is three and sportive to use Force has been closer to two be somewhere around here just so you can see them and we're going to draw her lips. So this is our lips and and we have completed the problem


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