In Exercises 1–18, graph each ellipse and locate the foci. x^2/16...

Question

In Exercises 1–18, graph each ellipse and locate the foci. x^2/16 +y^2/4 = 1

Accepted Answer

Hey everyone in this problem. We are asked to graph the following lips and determine its folk. I and the equation we're given is x squared over 36 plus Y squared over nine is equal to one. The first thing we want to do when we're given an equation for an ellipse. And were asked to graph it is rewrite that equation into standard form and recall that the standard form is going to be x minus h squared over a squared plus y minus K squared over B squared is equal to one. Now when we have it written in this form, it's gonna give us some important pieces of information that help us with our graph. So the center of our ellipse is going to be given by the coordinates. H comma K. The horizontal radius, it's gonna be given by A and the vertical radius is going to be given by B. So let's take our equation and try to write it in this form. We have x squared over 36 plus y squared over nine is equal to one. Now this is really close to standard form. All we need to do is write the denominators as something squared. So we have x squared over 36. We know that 36 can be written as six squared. So we can write this as x squared over six squared. And then we have plus y squared over night. We know that nine can be written as three squared. So we get y squared over three squared and all of this is equal to one. Now this is in standard form and we can pick out those important pieces of information so we have no nothing adding or subtracting to this X and Y term before we square them. And so are H and K. Values are going to be zero. That tells us that the center is going to be a zero and then our horizontal and vertical radius is are going to be given by this A. And B. Value. So let's write this all out. The center of our lips Is going to be at 00. Are horizontal radius is going to be given by A. Which is six. It's the square word of the denominator associated with the X squared term. And then our vertical radius is going to be three. It's our B term and it is the square root of the denominator associated with the y squared term. Now that we have our center and our horizontal and vertical radius we can draw our lips that's all the information we need. And so we have our X axis and our Y axis. We're just adding our units here. Alright so what we do is we started the center 00 and our horizontal radius is going to be the distance from the center to the edge of our lips. So we're gonna start at this center, we're going to move in the positive X direction. A total of six units. 123456. We have a point here We know this is a .60. That's going to be the edge of our lips. We go back to the center and do the same in the negative extraction here. We get this point This is a -60. Now we do the same in the vertical are vertical radius is three. So we're going from the center up three units and from the center down three units. So we get the following two points zero comma three and zero comma negative three. And if we connect these four points that we've drawn this is gonna give us our ellipse. All right, So this is what our lips looks like a rough sketch of it. We've got our graph. What we have to do now is find the foca. The folks are gonna 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. 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 we choose on our lips. Those are going to be our focus. Now the focus is going to lie on the major axis. The major axis in this case is going to be the horizontal axis because A is larger than B. Okay, so A equals six is larger than B equals three. Which tells us that the horizontal access is going to be the major taxes. So our folk I are gonna lie along our horizontal axis. And since our center is at 00, the horizontal axis is just the X axis. So our folks are going to lie along the X axis, know the coordinates for the focus. They're going to be equal distance from the center along the X axis. And so they're going to be located at zero plus or minus some distance F. In the extraction, they're going to be located at a y coordinate of zero because they're along that axis. So how do we find that value of F? What's that value of F. Well, recall that the value F is given to the following, F squared is going to be equal to a squared minus B squared. And that equations in this case because A is larger than be if you had it the other way around, you would flip the equation to B. B squared minus a squared. So we always want to take the major axis first, which we know is the horizontal axis. So we take our horizontal radius first. So we have x squared is equal to six squared minus three squared. So f squared is equal to 36 -9. We're gonna continue up here. That tells us that f squared is equal to 27. And so F taking the square root is plus or minus the square root of 27. Now, we want to see if we can simplify this at all. We want to see if we can find factors of 27 that are perfect squares. Now we know we can write 27 as nine times three and we know that nine is a perfect square. Using our square rules, we can write this as plus or minus. The square root of nine times the square root of three. Square root of nine is three. So we write this as plus or minus three times the square root of three. And so we can write our folk I plus or minus three root three comma zero. That is going to be the coordinates representing those two points. We have our graph, we have our folk. I let's look and see which answer choice is correct. So in answer D we see that the ellipse does not look like the ellipse. We drew, The shape goes the other direction. Okay, this has a bigger vertical radius than horizontal radius, which is not what we have in C. The ellipse looks like the right shape, but the points given don't match the points that we found. Okay, so it's not going to be C. What about B B looks like a similar shape. Let's take a look at the points. 03600, negative three negative 60. Those match the points on our lips. And so this is the same graph. And then if we look at the folk, I negative three root 30 and three root 30, those are the exact same folk I that we found. And so our answer for this one is going to be be that matches the folk I. And the graph that we found. Thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 1–18, graph each ellipse and locate the foci. x^2/16 +y^2/4 = 1

Key Concepts

Ellipse Definition

Graphing Ellipses

Foci of an Ellipse

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