In Exercises 19–24, find the standard form of the equation of eac...

Question

In Exercises 19–24, find the standard form of the equation of each ellipse and give the location of its foci.

Accepted Answer

Hey everyone in this problem. The graph of an ellipse is given below. Were asked to find its standard equation and the coordinates of the foca. Now when we're asked for the standard equation recall, the standard equation of an ellipse is given by x minus h squared over A squared plus y minus k squared over B squared is equal to one. Now H K. A. And B all give us important information about our lips. So H. K. The coordinate is going to represent the center of our ellipse. The A value is going to give the horizontal radius and the B value is going to give the vertical radius. So if we can find a church, K. A. And B. Then we'll have our standard equation. So we need to find the center, the horizontal radius and the vertical radius of this ellipse from our grasp. So we can look at this graph and from this we can see that right in the middle here is the center of our lips and that is at the origin. So the center of our lips is going to be at 00. The origin Which tells us that H is equal to zero and K. is equal to zero. What about the horizontal radius? So the horizontal radius is going to be the distance from the center to this outermost point on the right hand side, or to the outermost point on the left hand side? The radius and the horizontal direction. So because we're starting with our center at the origin, we can just look at the X value. We're going from an X. Value of zero up to an X. Value of five on this point on the right hand side. And so our horizontal radius is going to be five Which tells us that our a. value is equal to five. Now if the center is not at the origin it can get a little bit more confusing when you're looking for the horizontal radius. What you can do is find the distance from the right most point to the left most point. So in this case we have an X. Value of five. We subtract that from an X. Value of negative five. We have five minus negative five gives us 10. Okay and then we divide by 2. 10 is the entire diameter. We're looking for half of that. We divide by two and we get five. Okay So when our center is at zero we can do it the way we showed um it's simple to do it that way. When the center moves from the origin. If it gets a little bit confusing you can always find the damn it our first and divide by two. Either way will work. So we've done our horizontal radius. We're gonna do the same for the vertical radius. So we're starting at the center at a Y. Value of zero. We're going up to a Y. Value of three or down to a Y. Value of negative three. So that is a radius of three. So the vertical radius is going to be three, Which tells us that our B value is equal to three. Now we've found H K A and B. So we can put these into our standard form of our equation and write it out for this particular ellipse. So we're going to have x minus h. So x minus zero squared Divided by a squared divided by five squared plus why minus K squared? So why minus zero squared divided by b squared? three Squared is equal to what? And now we can simplify this. So we have X squared over 25 plus Y squared over nine is equal to one. And that is our standard equation for our lips. And that is the first part of the problem. So now we want to move to the second part of the problem and we're asked to find the coordinates of the foca. The folks are going to be two points whose sum of distances from any point on the ellipse is always the same. So what that means is we're gonna find two points in the distance from the first point to a point P on the ellipse. Plus the distance from the second point to that point P on the ellipse is going to give some value and that value is going to be the same no matter which point P you choose along that ellipse. All right now the folk, I always lie on the major axis, which accesses the major axis in this case. Well, the horizontal radius is larger than the vertical radius. That tells us that the horizontal axis is the major axis. So we're talking the horizontal axis of our lips now because our lips is centered at 00. The horizontal axis just corresponds with the X axis. So these are gonna ally along the X axis and they're going to be an equal distance from the center. So if they're on the X axis, we're going to start at our center which has an X coordinate of zero. And they're going to be an equal distance away from that. So we're gonna say that plus or minus some distance F. And there on the X axis. So we know that the y value is just going to be zero. So the coordinates of our folks are going to be given by this zero plus or minus f comma zero. Now we need to calculate f. How do we do that? Well, recall that F squared is going to be given by a squared minus b squared because we have a horizontal major axis. We do a squared minus b squared. If you have a vertical access being your major access, you would do b squared minus a squared. So you always take the bigger one first. All right. So in our case a squared minus B squared we get f squared is equal to five squared minus three squared. Okay, remember that represents the horizontal radius and B represents a vertical radius. This gives us F squared is equal to 25 minus nine which is equal to 16. And taking the square root F is going to be plus or minus four. And so the coordinates of our folks are gonna be at plus or minus four and zero. So if we look at the answer traces, we found our standard equation of our lips, we found the coordinates of our focus and we see that answer choice A X squared over 25 plus Y squared over nine equals one. That matches the equation we found and the folk I. R. At negative +40 and +40, which also match the coordinates that we found. And so A is going to be the correct answer choice here. Thanks everyone for watching. I hope this video helped see you in the next one.

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In Exercises 19–24, find the standard form of the equation of each ellipse and give the location of its foci.