In Exercises 33–42, use the center, vertices, and asymptotes to g...

Question

In Exercises 33–42, use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x+4)^2/9−(y+3)^2/16=1

Accepted Answer

Hello, today we're gonna be graphing the equation of the hyperbola. Then we're gonna find the equations for the diagnose totes and the coordinates of the ci. So what we are given is X minus two squared over minus Y minus four squared over 49 is equal to one. This is given to us in the standard form of the equation of the hyperbola. But there's a few things to note about this equation. First, the X and Y values are given to us as binomials. And that tells us that the equation of the hyperbole is not located at the origin. The center is going to be defined as H comma K. And another thing to note is that since the X term is the leading term on the left hand side, the major axis for the hyperbola is going to be the X axis. So since we know that this is not located at the origin and the major axis is the X axis, the general form of the equation of the hyperbola is given to us as X minus H squared over A squared minus Y minus K squared over B squared is equal to one, this is the equation of a hyperbola where the hyperbola opens up to the left and to the right. The distance of the vertices from the center is defined as some distance A and the height of the ascent total box is defined as some distance B from the center. So let's go ahead and use the general form of the equation of the hyperbola to get our A and B values and to get the center. So we know that 25 is in the A squid spot and we know that 49 is in the B squirt spot. So we can go ahead and set A squared equal to 25 B squared equal to 49. If we take the square root of both of the equations, we get A is equal to plus or minus five and B is equal to plus or minus seven. Now let's go ahead and get the center to get the H value. We're going to take a look at the X quantity we have X minus two squared. This is in the general form of X minus H squared and this is where H is equal to two. And for the K value, we're going to take a look at the Y value, we are given Y minus four squared. This is in the general form of Y minus K squared. And this is where K is equal to four. And that means the center of the hyperbola is going to be defined at two comma four. So now that we have the center of the hyperbola and the A and B values, we can use this information to sketch the hyperbola. So we know that the hyperbole is located at two comma four. The vertices are gonna be located a distance of five to the right and to the left of the center of the hyperbola, that means the vertices are gonna be located at seven comma four and negative three comma four. And the acid total box is going to have a height of seven units above and below the center of the hyperbola. So that means the height is going to be defined by two comma and two comma negative three. We can connect these four dots by drawing our as in total box and we can use the ascent total box to create a rough sketch of the diagonal ay totes. Now keep in mind that this hyperbola opens up to the right and to the left. So if we draw the hyperbola with respect to the ascent totes, the hyperbola is going to have a rough shape like this. So now that we have the general shape of the hyperbola, the next thing to do is to find the locations of the, now the are going to be some distance C to the right and to the left of the hyperbola. So we need to find this distance of C for the hyperbola, we have an equation to help us get that. We have C squared is equal to A squared plus B squared. And we know what our A squared and B squared values are going to be A squared is 25 B squared is 49. So if we plug this into the equation, we get C squared is equal to 25 plus 25 plus 49 is going to give us 74. So we have C squared is equal to 74. And if we take the square root of both sides of the equation, we get C is equal to plus or minus the square root of 74. So what this means is that the are gonna be located a distance of square root 74 to the right and to the left of the center of the hyperbola. And what that means is that we're going to be adding subtracting the square of to the X value of the center of the hyperbola. This means that the locations of the are going to be at two plus square root 74 comma four and two minus the square root of comma four. So this is going to be the locations of the sign. Now, we need to find the equations for the diagonal asy totes. Now recall from earlier, we said that this hyperbola is not located at the origin and the major axis is the X axis. That means the general form of the equations of the diagonal asy totes is given to us as Y is equal to plus or minus B over A times the quantity X minus H plus K. We know what our B and A values are B is equal to seven and A is equal to five. We also know that H is equal to two and K is equal to four. So if we plug this into the general form of the equations of the Diago Asy totes, we end up getting two equations. Y is equal to positive 7/ times the quantity X minus H which is X minus two plus K which is four. And we also get Y is equal to negative 7/5 times the quantity X minus two plus four. On the left hand side, we're going to distribute 7/5 into X minus two. And that is going to give us Y is equal to 7/5, X minus 7/5 times two, which is 14/5 plus four. And on the right hand side, we're going to distribute negative 7/5 into X minus two. That's going to give us Y is equal to negative 7/5, X plus 14/5 plus four. Now, on the left hand side, if we add negative 14/5 with four, we get the value of 6/5. So the first equation is going to be Y is equal to 7/ X plus 6/5. And on the right hand side, we're going to add 14/5 with four and that's going to give us 34/5. So the second equation for the diagonal ascent totes is going to be negative 7/5 X plus 34/5. So just to recap, we just found the equations of the diagonal ascent totes. We have the locations of the and we have the rough sketch of the hyperbola with the center and asy totes located uh center and vertices labeled. And with that being said, this is going to be the answer to this problem. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

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In Exercises 33–42, use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x+4)^2/9−(y+3)^2/16=1

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