Graph the ellipse and locate the foci. 9x^2 + 4y^2 - 18x + 8y -23...

Question

Graph the ellipse and locate the foci. 9x^2 + 4y^2 - 18x + 8y -23 = 0

Accepted Answer

Hello. Today we're going to be using the given equation to sketch the graph of the ellipse. So what we are given is four X squared plus 25 Y squared plus 24 X minus 50 Y minus 39 is equal to zero. Now in order to sketch the graph of the ellipse, we're going to need to first rewrite this given equation in standard form. In order to do that. Let's go ahead and group up all of our X terms together all the white terms together and move any constance to the right hand side of the equation. Doing this is going to give us four, X squared plus 24 X plus 25 Y squared minus 50 Y Is going to equal to 39. Now what we can go ahead and do is factor out a common factor in both of our groups on the left hand group, both four and 24 are divisible by four. So we can go ahead and factor out of four. And for the wide group both 25 and 50 are both divisible by 25. So we can factor out at 25. So factoring out the common factors is going to give us four times the quantity X squared plus six, X plus 25 times the quantity y squared minus two Y is equal to 39. Now, what we can go ahead and do is complete the square on both the X and Y groups. So this way we can go ahead and rewrite this in standard form. So completing the square on the X group is going to give us the third term C is equal to six divided by two quantity squared and that is going to equal to nine. And completing the square on the Y group is going to give us negative two divided by two squared, which is going to equal to one. So now we're gonna go ahead and add these quantities into our equation and doing this is going to give us four times X squared plus six, X plus nine plus 25 times y squared minus two, Y plus one Is equal to 39. Now we need to be careful here because since we're introducing the values of nine and one to the left hand side of the equation, we must also do the same to the right hand side, but we did factor out a common factor of four and 25 from the groups before completing the square. So if we wanted to add nine to the right hand side of the equation, we will have to add nine times four. So that's what we're going to need to do in order to balance the equation and the same thing goes for the Y value since we're adding a one, but we factor out 25, we're also going to need to add 25 times one to the right hand side of the equation and 25 times one is just going to be 25. So balancing the equation what we go ahead and have on the right hand side is 100 because 39 plus nine times four plus 25 is going to give us 100. And on the left hand side we can simplify the to try no meals since they are now both factory ble x squared plus six, X plus nine is going to factor two, X plus three squared. So what we have is four times X plus three squared. And that's gonna be plus 25 times y squared minus two, Y plus one. Factoring that is going to give us y minus one squared. So we have 25 times y minus one squared is equal to 100. Now, finally, we want to go ahead and set the equation equal to one. In order to do that. We're going to divide every term in this equation by 100 that's going to give us X plus three squared over 25 plus y minus one squared over four is equal to one. And this is going to be the standard form of the equation of the ellipse. So this is an equation of the ellipse where the center of the ellipse is not at the origin and notice that the 25 under the X term is greater than the denominator. Under the Y term. With that being said, the standard form of this ellipse is going to be x minus h squared over a squared plus y minus k squared over, B squared is equal to one. And this is where the transverse axis is going to be the x axis. So we're going to have a horizontal ellipse. So let's go ahead and figure this out by identifying the center of the ellipse. The center is given to us as a church comma K. And we can get these H and K values by looking at the X and Y values. See here, X plus three is not in the standard form of x minus H but we can rewrite it to be x minus negative three. This is where H is going to equal to -3. And if we take a look at the y value, we have y minus one, which is in the standard form of y minus K. So this is where K is equal to positive one. So that means our center of the ellipse is going to be located and negative three comma one. So now that we have the center of the ellipse, we need to go ahead and identify the major and minor vertex is for the major vergis is The major versaces are denoted by the value of a and here a squared is equal to 25. Taking the square root of both sides is going to give us a is equal to plus or - and the major axis lies on the transverse axis, which is going to be the X axis. However keep in mind that our center of the ellipse is not at the origin, it's at negative three comma one. So that means that we're going to add positive and negative 52 negative three in our center. So the vertex is are going to be negative three plus five comma one and negative three minus five comma one. And this is going to give us the vertex is of 2: and negative eight comma one. So this is going to be the location of the major vert is's Now we need to figure out the location of the minor versaces and the minor versaces are denoted by the B variable Here, b squared is equal to four. And taking the square root of both sides is going to give us B is equal to plus or -2. So what we need to do now is go ahead and add two and negative two to the y value of the center. Since the minor vergis is are going to be located on the y axis. So that's gonna give us the Virtus is negative three comma one plus two and negative three comma one minus two. And this is going to give us the minor versus is of negative three comma three and negative three comma negative one. And now we just need to identify the focus i the equation of the faux side is given to us for ellipses as C is equal to the square root of a squared minus b squared a squared was 25 b squared was four. So what we have is the square root of 25 minus four. And that is going to equal to the square root of 21. The square root of 21 cannot be simplified any further. So we're going to leave it as plus or -21 and the fossa are located along the major axis of the of the ellipse. So what we're going to do is add this to the x values of the center of the ellipse. So we're going to have three plus the square root of comma one and the location of the other. Other I'm sorry negative three plus square root 21 comma one and the other folks is going to be negative three minus the square root of 21 comma one. And these are going to be the faucet for the ellipse. Now that we've identified all the information we need we just need to select the ellipse that accurately represents all these values. And this ellipse is going to be option D. C. Option D has the center of negative three comma one and the vertex is on the major axis are located at two comma one and negative eight comma one For the minor versaces we have negative 3:03 A negative three comma negative one. And the focus I are given to us as negative three minus the square root of 21 comma one and negative three plus the square root of 21 comma one. So that means that 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 an ellipse and I'll go ahead and see you all in the next video.

Graph the ellipse and locate the foci. 9x^2 + 4y^2 - 18x + 8y -23 = 0

Key Concepts

Ellipse Standard Form

Completing the Square

Foci of an Ellipse

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