Graph the hyperbola. Locate the foci and find the equations of th...

Question

Graph the hyperbola. Locate the foci and find the equations of the asymptotes. ((x-2)^2)/25 - ((y+3)^2)/16 = 1

Accepted Answer

Hello today we're going to be using the given equation to identify the graph of the hyperbole to. So what we are given is X plus one quantity squared Over 36 -6 -2 quantity squared over 64 is equal to one. Now this is the standard form of the equation of the hyperbole to that's not located at the origin. So the standard form is given to us as x minus H squared over a squared minus y minus k squared over B squared is equal to one. And this is the equation of a hyperbole that opens up to the left and right but is not located at the origin. So before we start anything, let's go ahead and identify the center of this hyperbole to. So the center is given to us in the form of h comma K. And we can get our H and K values by looking at the X and Y quantities for the X quantity we are given X plus one. But this is not in the standard form of x minus H. So we can go ahead and rewrite this to give us x minus negative one. And this is in the standard form of x minus H where H is equal to negative one. And now let's go ahead and identify our k variable. We are given why -2. And this is in the standard form of Y -K. This is where K is equal to positive two. So the center of this hyperbole to is going to be located at negative one comma two. Now let's go ahead and identify A and B values because we're going to need that to identify the vertex is fosse and assume totes. So currently we have A squared is equal to b squared is equal to 64. If we take the square root of the a equation we get a is equal to plus or -6 and if we take the square root of the B equation we get B is equal to plus or -8. So now that we have these values, let's go ahead and first start by identifying the diagonal assim totes. So the equation of the diagonal totes is given to us as why minus K. Is equal to plus or minus B over a times the quantity x minus H. This is the equation of the ASM tote where the x axis is the transverse axis and is not located at the origin. So we know what our H and K values are, it's negative one and two and we know what our B and A values are which is eight and six. So we're just gonna go ahead and plug in these values into this equation and this is going to give us y minus two is equal to plus or minus B over A which is 8/6 times the quantity x minus H which is x minus negative one which is the equivalent of X plus one. We can simplify 8/6 to 4/3. So the equations of diagonal assim totes are going to be y minus two is equal to plus or minus 4/3 times the quantity X plus one. That's the first piece of information we need for the hyperbole to next. Let's go ahead and identify the vertex is now because the transport access is with respect to the X axis, we're going to be adding a certain distance to the current center. See A represents the distance of the Aston total box to the left and right of the center of the hyperbole to So in order to get the vertex is we're going to be adding and subtracting six to the X value of the center of the hyperbole to and that's going to give us negative one plus six comma two and negative one minus six comma two. And this is going to give us the vertex is located at five comma two and negative seven comma two. Now that we have the vertex is we need to go ahead and identify the fosse. Now the equation of the fosse is given to us as c is equal to the square root of a squared plus b squared. So plugging in the known a squared and b squared values is going to give us the square root of 36 plus 64 which is going to simplify to the square root of 100 which evaluates to plus or minus 10. Now the focus, I are going to be located with respect to the transfers axis which is the X axis. Since the location of the fosse is 10 units to the right and left of the center we're going to be adding and subtracting 10 to the X values of the center of the hyperbole. And that's going to give us negative one plus 10 comma two and negative one minus 10 comma two. And that's going to give us the location of the fosse Which is going to be 9:02 and -11:02. So now that we have the fosse versaces and as I'm totes let's go ahead and choose the option that accurately represents all this information. And that option is going to be D d tells us that the ascent toads are y minus two is equal to plus or minus 4/3 times X plus one. Which is what we've identified up here. It also tells us that the focus I are located at negative 11 comma two and nine comma two which you've identified here and the vertex is are located at negative seven comma two and five comma two which we've identified up here. So with that being said, the answer to this problem is going to be D. So I hope this video helps you in understanding how to identify the graph of the hyperbole and I'll go ahead and see you all in the next video

Graph the hyperbola. Locate the foci and find the equations of the asymptotes. ((x-2)^2)/25 - ((y+3)^2)/16 = 1

Key Concepts

Hyperbola Definition

Foci of a Hyperbola

Asymptotes of a Hyperbola

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