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Ahyperbola has asymptotes of y 3and y The vertices are 16 units apart and lie on the same vertical line Find the center of the hyperbola, the distance from the cent...

Question

Ahyperbola has asymptotes of y 3and y The vertices are 16 units apart and lie on the same vertical line Find the center of the hyperbola, the distance from the center to the focus_ Its eccentricity: andThe center of the hyperbolaThe distance from the focus to the centerThe eccentricity isSubmit Qucstion

Ahyperbola has asymptotes of y 3and y The vertices are 16 units apart and lie on the same vertical line Find the center of the hyperbola, the distance from the center to the focus_ Its eccentricity: and The center of the hyperbola The distance from the focus to the center The eccentricity is Submit Qucstion



Answers

Find the center, foci, vertices, asymptotes, and eccentricity of the given hyperbola. Graph the hyperbola. $$ 9 x^{2}-16 y^{2}+144=0 $$

The center of this, hyper Bella is 3 1/4. Its furnaces are three 1/4 plus two because of this for right here. So two and 1/4 And 3 1 4th -2. 1/4 -2 So -1 and 3/4. Uh C square. It is 13. So the focus I. R. Three And 1/4 plus and minus the square to 13. He asked. Some toads are why -1 4th equals plus a minus two thirds. X minus three, Eccentricity Square to 13. Over to. All right, go to the center which is 343, 1/4. I mean. Okay, so we'll put it right here go out three in the X direction. So three horizontally. Both ways here here go up to mhm. Down to drive a rectangle through those corn or through those points, draw lines through the center that go through the corners. Those are the accent oates. Okay, since why is first the vertex is here. Uh we're supposed to be three. The vortices are supposed to be three. Yes and two and a 4th And -1 in 3. 4 6. Get close enough. All right then it opens like this so it stays close to the rectangle and goes towards the sm totes And then the photos are 1/4 plus and minus the square to 13. Um the square root of 13 is more than three but not quite four. So let's just save 3 1/2. So three about 3 75 and 3.25. Oh no -3.25. So 123. So we're here and right there.

So the key to this problem is being able to complete the square. Uh And what you will also notice is that X squared is by itself, that's why squared Plus 36 wide minus 72 is equal to zero. So what I would do this is because every teacher is a little different is I would first ad 72 to the right side and I would factor out a negative nine from both the Y squared And the 36. Uh And you can double check that this is equal. If I were to distribute that negative nine in there, it would equal what's right there and then I leave a space right here because I'm going to complete the square, which is taking half of this number negative for squared is equal to positive for. So that's what I'm going to add right here, but I'm not really adding forward to the problem, I am subtracting -910. I guess I say adding -9 times four will be 36. So if I remove 36 from the left side, they also have to remove 36 from the right side. So as I'm looking at that problems only to -36 is 36. And what I have on the left side is nine times a perfect square china meal. So it's why minus two squared, I'm almost there. What I need to do is get this equation equal to 36 which I can do by dividing every piece by 36. And it might even help you to rewrite this as x minus zero squared over six squared And then nine goes into 36 4 times. But I'm already has two squared. And you can double check that this math is correct. But the reason why I go through this is because now I can quickly look at this and say my center. Is that the opposite of these two numbers? So it's at 02. Well um I'll give you the graph so shifted up to is my center and then my A value needs to be six. It's a horizontal axis because the X term is the positive. So to the right six 123456 is one vertex and to the left The other vertex list them for you plus or -6. Why couldn't stay the same? Might be value would be too. So I am a big fan of making a box going around and then making your ass in totes go upon through the corner of those boxes. And then you can say, okay well my asientos are clearly Y equals. Instead of saying up to write six, you can reduce that to 1/3 And then X -0 0 Plus The Y Coordinate of two. Um I think we have enough to create the graph. The only thing we're missing though is the folk. I and if you follow the Pythagorean theorem, a squared plus B squared equals c squared, I think it's pretty clear that 36 plus four is equal to see square, so that's 40, but you need to square root that, You know, so that's four times 10. So that's the folk. I if I rewrite it and simplest right to perform is to route 10 and up to still. I think that's everything I need. So you had the graph kind of cut off my ass and talks. We had a folk. I the Asientos Center advertises all right there.

Okay, center of this. Parabola is 04. Um The vertex is our uh okay be a six here. So 04 plus six and zero for minus six C squared is a squared plus B squared. So it's 37. And the first I then are zero For plus and minus the square to 37. Then they asked him towards our Y -4 equals Plus -6 times x. The Eccentricity 37/6. Alright so go to the center which is positive for zero. Okay go out one on the horizontal each way Go up and down. 6 4 456 123456 Drop boxes. Draw a box through those four points. Okay dry rectangle. Okay. Draw lines that go through the center and the corners of the box. Oh my gosh okay. These are the asem totes. Okay because why is first then we know it's going to open on the Y so this is this should be the vertex which is 10 and then here's the other one minus two and then it's right in between the assam totes. Okay so the hyperbole is only the red part. The rest was to draw. Okay the first ir at four plus or minus the square root 2 37 which is about four plus or minus six. So 10 and negative to just a little bit past the vertex. Just barely and that makes sense because it's so skinny

I know that the problem just as X squared. But I'd like to add in that minus zero. That helps me. Uh and the same thing with why I know it just says why square but it helps me to see uh zero in there. Yeah because that tells me the location of the center is at zero. I should be underlying the one with the experts. So 00 is the center. So as I look at this hyper below, I already know that were centered at 00. The next thing that I need to point out is the X term is positive. So where are transverse axis is moving left and right. Um And how many units left and right is based off of this number. That's your A value. Uh Well sorry, a squared equals nine. So when you square root that the A value is left and right. Three units from the center left and right because of the X. Right there. So my advertises I'm just going to write plus or minus three Common zero because the Y coordinate has not changed at all. So let me enable a few things or text one vertex to in the center. And then over here We have 16 which is your b squared value. So what I typically do. So B equals for is I'll make a box Going up four and down for. And uh the bird, the box includes the vertex is but the reason why I make this box is because the asem totes go diagonally through that box. And that helps me come up with the equation of the ascent oats. Because as you look closely at that, then it's the equation Y equals plus or minus because we have a positive slope in a negative slope up four, right, three X. And then I could write plus zero because it goes to the origin but that should be fine. So we had the center with advertises, the only thing we're missing now is the focus or folk. I, and that's going to be using the formula A squared plus B squared equals C squared. Well, we already know that A squared is nine, B squared is 16, so 25 when you add that together. So the C value is five. So that means the folk I are both located um Plus or -5 units from the center along the X axis. So right here would be one focus and over here would be the second focus and that's our graph, so many circle all of this in green because those are all the answers I believe we're supposed to have. Now let's move on.


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