In Exercises 1–18, graph each ellipse and locate the foci. x^2/49...

Question

In Exercises 1–18, graph each ellipse and locate the foci. x^2/49 +y^2/81 = 1

Accepted Answer

Hey everyone welcome back in this problem. We are asked to graph the following ellipse and determine its folk. I were given X squared over 36 plus Y squared over 100 is equal to one. Now, when we're given an equation for an ellipse and we want to graph it, we want to put it into the standard form of the equation first and recall that the standard form 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 we want to put it into this form because it gives us important information about our ellipse. When it's in this form, we can tell that the center of our lips is that the coordinate H comma K. Our horizontal radius is given by a and our vertical radius is given by me. So for our equation X squared over plus Y squared over 100 is equal to one. This is already very close to being in standard form. The only thing we need to do is take these denominators and write them as something squared. So to do that, we have x squared over 36. We can write 36 as six squared. So we have x squared over six squared and then we have y squared over 100. We can write 100 as 10 squared. So we have y squared over 10 squared and this is all equal to one. Alright, so now we have our standard equation of our lips. Let's figure out those important values that we talked about being able to find. So our X and Y terms don't have anything added or subtracted from them before they're squared. And so our agent K values are just going to be zero, which tells us that the center of our lips is going to be at the origin 00. We know that our horizontal radius is going to be given by our a value which in this case is six and that are vertical radius is given by our b value, which is 10. Now with the center of our lips are horizontal and vertical radius, we can go ahead and graph it. We don't need any more information. So let's go ahead and do that. We draw our axes, we have our X and y axes In each of these ticks we're gonna say is two units. We're gonna do the same along the other directions. All right now, in order to draw this, what we want to do is start at the center which is at the origin. We're gonna move in the horizontal direction, we're gonna start in the positive horizontal direction. We're gonna move our horizontal radius of six units. So we move six units in this direction, we draw a point. That point is at six comma zero and then we do the same in the negative direction. So we started our center 00. We move in the negative horizontal direction. Our Radius six units. Let me draw another point. And that point in this case is -60. And we do the same for the vertical. So we move in the positive vertical direction. So upwards 10 units draw point. That point is at 010 and then in the negative direction Same thing 10 units. And that point is going to be at zero and negative 10. And now we can connect these four points and this is gonna give us our lips, it's gonna look something like this. Alright so we have our graph now we need to find the folk. I and the folks are gonna be two points whose sum of distances from any point on the ellipse is always the same. And what do I mean by that? We're gonna pick two points or we're gonna find two points inside of our lips. The distance from the first point to a point P on her lips plus the distance from the second point to that point P on our lips is gonna give us some value. And no matter which point P on our lips we choose that value is going to be the same. Alright so our focus is gonna lie along our major axis. What's our major axis? Well our vertical radius is larger than our horizontal radius. And so the vertical axis is going to be our major axis. Were centered at the center or at the origin. And so the y axis is going to be our major axis. So our folks are going to be located at an x value of zero and then the Y value what we're going to start at the center and they're going to be equally spaced from the center. So they're going to be at zero plus or minus some distance F from that center in the vertical direction. What's the value of F. While we can calculate that and recall the f squared is given by B squared minus a squared because b squared is larger. If you get it, the situation where a squared is larger, you would do the opposite. But in this case b squared is larger. So we have b squared minus a squared. So we're gonna have f squared is equal to 10 squared minus six squared, Which is equal to 100 -36, Which gives us an f squared value of 64. And if we take the square root we get that F is going to be equal to plus or minus eight. And so our folk, I are going to be located at zero comma plus or minus eight And we can draw those on our graph if we like 0 -8 is going to be here and we're going to draw them in blue and in the positive direction these other foca is going to be up here in blue. Okay, along that vertical axis. So if we look at our answer choices, we see indeed that the shape of the ellipse looks good. It looks similar to what we drew. We have the 0.0 10 600, negative 10 and negative 60, which match the points that we had in our lips. And we have the folk I at zero and negative eight and zero and eight, which is what we found as well. And so D. Is going to be the correct answer. Thanks everyone for watching. I hope this video helped you in the next one.

In Exercises 1–18, graph each ellipse and locate the foci. x^2/49 +y^2/81 = 1

Key Concepts

Ellipse Definition

Graphing Ellipses

Foci of an Ellipse

Watch next