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Solve the given problems.Find the equation of the hyperbola with asymptotes $x-y=-1$ and $ar{x}+y=-3$ and vertex (3,-1)...

Question

Solve the given problems.Find the equation of the hyperbola with asymptotes $x-y=-1$ and $ar{x}+y=-3$ and vertex (3,-1)

Solve the given problems. Find the equation of the hyperbola with asymptotes $x-y=-1$ and $\bar{x}+y=-3$ and vertex (3,-1)



Answers

Find an equation for the conic that satisfies the given conditions.

Hyperbola, vertices $ (\pm3, 0) $, asymptotes $ y = \pm2x $

So in this problem, we're asked to put this equation for hyperbole up in the standard form. Could find its ask them, toots, and to grab bag purple. So the first thing to do so to put it in standard for we want this number over on this side to be like So what I'm going to do is someone going to divide this side by three. Which means I have to define this other side by three. So we get on one side y squared over three minus ex cleared. It's equal to what? So this will be Theo equation for the privilege in standard form. So just a quick note will be the fact that we have the term with why first means that hurt Hyperba is going to open up and done. And so the folks are also going to be on the y axis. So, um, let's continue to figure it out to the as kids. So this is thea Question in standard. Um, we said that bull guy are in the UAE accents, so it's going to be going toe, have co ordinate zero and plus and minus C and sees a find like this we'll get to that leader and aspic. Essen toads. They earned the two lines with equations. Why equal two plus and minus a over B? Again, this is from what we get from the equation. Time, Sex. All right, so first thing we would like to do here, let's figure out who x squared and be screwed. Are we do that from looking at you creation. So I'm first time the new morita. This is going to be a squared. So there you go. And B squared is the denominator of the second term. At our case, this is the same. Is writing x squared over one soapy squid will be one, and we get A and B bye. Um, applying square root. Right? So, um, for a we have a square root of three which does not simplify. And for beauty at the square root of fine, which is also when, Right. So now we can figure out to this value for seat iss that will be square root of a squared plus B squared that it's three coastline. It is square with a four, which is to okay. And so now we have our the co ordinates for the food and pretty asking coats. We just need to plug in the values for A and B, So this is going to be equal to plus and minus. And that is, um, square root of three. Over be bliss one. I'm sick. So we could also write this without the baby, since it's just one. So we judge Wright Square root of three times. Thanks. So this will be the attitudes. Why equal to this is the, um are the okay equations, for the s intents. So, no, we can get to graphing. So we have first pretty the values for a We need the full kind we know sees he bolted to and we have this equation. So, um, you so? And it's equal to square root of three. Okay, are See you two and the what? Essen totes. I'm going to name the s in tune with the positive, uh, slope. I'm going to need a white one. So what I want it's going to be equal to square did three times x and Y two. I'm gonna name it the one with the negative slip. Still be negative. Square root of three times six. Okay, so um, we know because first term missed a lot with why that the white access is the local accents. So first, I'm going Teoh right down the value for a gonna find it in the Y axis in the States over here and minus a hey, so the full guy are going to be over here. Coordinates zero plus two is that's the one and at too, the one with negative sea or in this case, negative, too. That should be around here and now to grab the let's first grab the actual, um, hi, purple. So since this is a local accent, they're going to open to the to the They're going to open up and down and they're going to cross the white access at the values of A and minus A. So it's going to come around here, cross it square root of three, and grow up. In this case, it's going to end down cross at Negative Square would go there. Right. So this is the grab for the actual hit her Bob. And finally, why one of y two. Those are the Essen toots. Their going to cross the origin. So one of them is going to go around here. So since this has be positive slope, this is going to be why one and 2nd 1 It's also going to crust. Uh, origin should go somewhere around here somewhere over here. And since it has negative slope, this should be eaten white, too.

So I have to find the equation of the acid Toepfer, the given hot parabola which is y squared over three square miles. Expert over three squared equals one. So the first thing I want to do is talk about when the aschen tote is so Nazem Took to occur is a straight line which the curve approaches without crossing. So if we go sufficiently far along the line, that curve becomes arbitrarily close. So let's take a look on how we can solve this of the hot the Assam totes of this function. Most equation. So the Assam toads are the lines by purple. Intends to be at that positive and negative. Finn be so foreign. Up, down high purple of the acid totes are ah, yielded with dysfunction. Why's he going to plus or minus a over B X minus a plus? Que This is our equation. So with this equation, we can go ahead and now take a look at our given equation and plug in these values. So were given were given the hyper Bella in the form of the standard equation for an up down facing my proble which is given this form why minus k squared over a squared minus K minus H squared Overby swears equals one. So now we can go ahead and just get the h k A B values from this equation. So if we do that, we get that a is equal to three. As you can see where I must say, I see that day is here and A is here so easy. But 03 b is also equal to three and a TSH comma. K Values are just zero and zero because there's no value here. There's no agent Cabe equated. So let's assume that this is negative zero there and X squared minus zero is just still I squared r y squared minus zero is still y squared. So our values for H comma K are equal to zero comma zero. So with his knowledge, we can now go ahead and, uh, find our hyperba. So the question access, what are the acid toe to the function? So let's go ahead and do this. So we say that why is equal to remember I'm just really in this equation, plus or minus three over three times X minus zero plus zero. Basically, I'm just plugging all these values here into this equation into this big equation here. And I get that wise you with plus or minus two over three X minus zero plus zero. If we go and signifies one more time to get that, why is he going to plus or minus one times X plus three over three is one This yields. Why's he going to plus or minus X? And this is our final answer for the ass in tow.

We were given this equation over here off a hyper Bella. Hi, Ebola. Oh, are asked to find the asset don'ts of a hyper boa whereas to find the Assam toes. And so we noticed over here that this is a parable and the horrors on the orientation cause the X the term that with the eggs is positive Something's a practiced it. Uh, this over here is positive. The term the expression containing the exit positive and therefore the hyper Bhola is in horse on the orientation. And so, through this rule over here tells us how to find the Assam totes of I broke hyperba with a horizontal orientation in central comma and notice. And it's really important that this is gonna be the rule if and only of the ass. The hyper boa is is one with a horizontal orientation in a center each key. Ah, horizontal orientation is extremely important. And basically what that means is if the X minus eight square term, if that's not being subtracted, if that's positive, we have so ah hyperba in the x o in the horizontal orientation and that. And that's why we can use this who we notice over here. That's the case. Uh, this over here is positive. That expression containing the access now being subtracted like this y over here. And so therefore, it's in horizontal orientation. It's really important to know that. And so now we can get to it. I remember the general form of a hyperba is gonna be heads off. That's minus X squared over a squared minus. Why? My has k squared over B square does he could want. And this isn't only the horse on toe orientation. Uh, is God s O. R a squared is the denominator of the X minus eight squared and the skirt is the denominator of the Y minus k square turn. And so what? We have to find out r b r a r h anarchy. So let's do that are a value is gonna be you. Whoa. Did they help us out over you there with that? The questions were in. It's actually helping us rather than writing five Square does 25 are solving the equation to find out what it is. It's writing five square. And therefore, since this isn't standard form, we know that the thing that's being scared squared underneath the X minus eight square terms are a valley. So therefore, five is our a value. Our be value similarly is gonna be too Are h value is gonna be the, uh, thing after the X minus r each values gonna be three and r k value is gonna be the thing after the y minus. So you might think it's why Post for How do I do that? Well, why Post four is gonna be the same thing as a Y minus negative four. Therefore, whatever is after the Y minus negative fours are key values. Okay, is equal to negative for. And so Oh, now that we have figured how everything let's substitute all the values into this so we know their ass in tow is interview y equals plus or minus B ovary be over a so two over five times X minus age s O X minus three plus or plus plus que. And we figured out that our K value was negative for so that is gonna be minus for so plus negative for is the same thing as minus for So therefore ask himto is going to be plus or minus two over five times X minus three, minus four. And the way we found out was were used this rule over here. But this rule over here, um, only on Lee works if the whore the ask himto if the hyperbole is in the horse on the orientation. And so I hope that cell phone, this is our ask him to

So as to find the equation of the acid totes for this hyperbolic equation. So first time, alas, Antonio is so innocent o to a curve is a straight line which the curve approaches without crossing. So with that knowledge, we can go ahead and solve for these Ah, yes, and Tony equations. So the equation for a nascent tote is yielded with this equation. Why is equal to plus or minus a over B times X minus h plus que beautiful. So for an up, down facing hyper Bella, this is the equation for the acid totes. So now we're gonna cocked it. The hyperbolic properties of the equation were given this equation right here. So if we go ahead and do this, it's already in the perfect form for us. So we have to do is read it properly. So we know the equation for the standard equation for hyperbole is given with this. Why, minus k squared over a squared minus K minus, h squared all over B squared yields one. So now that we know this, we can go ahead and solve for or not solved, just kind of extract all the values from this equation by following the rules set by this equation. So go ahead. Look at this. We get h Comma K is equal to something, eh? Is equal to something and be easy Lis something. And then we're gonna go ahead and take the values you get from here and plugged into this equation to get the Assam totes the equation. So we see, let's so first to start with a So I see a down here and this corresponding a value is three. So we've put in three for that. So be same goes, we get six h comma K. So this K value is negative. Three or three rather. And this h value is five. So we get the values off. Five, comma three. And the reason we don't use that negative. It has been the negatives in this equation. So we used the positive. So we have these values now. So now I wouldn't go heads plug these into this equation, which I'm gonna do in the next page. So it's gonna look something like this. Why is equal to plus or minus three over six times X minus five plus three. So if we go ahead and refine this a little bit. We simplify it, we get that. Why is equal to 1/2 plus or minus 1/2 X plus five plus three? And that was her answer for the equation of the acid totes.


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