In Exercises 1–18, graph each ellipse and locate the foci. x² = 1...

Question

In Exercises 1–18, graph each ellipse and locate the foci. x² = 1 – 4y²

Accepted Answer

Hey everyone welcome back in this problem. We are asked to graph the following lips and determine its folk. I and the equation we're given is y squared is equal to one minus X squared. Now, when we're given an equation for an ellipse morass to graphic what we want to do is rewrite that equation so that it's in standard form. Now recall that the standard form of the ellipse equation is x minus h squared over a squared plus y minus k squared over B squared is equal to one. Why do we want to write it like this? Well, we're able to pick out important pieces of information from the equation in this form. So the point H K. Is going to tell us the center of our lips, the value of A is going to be our horizontal radius and the value of B is going to be our vertical radius. So for our equation were given y squared is equal to one minus 16 X squared. And we want to write this in our standard form. So we want the X squared and y squared form or sorry, the X squared and y squared term together on one side of the equation and number one on the other side. So let's move the 16 X squared term over. We're gonna have 16 X squared plus y squared is equal to one. Now we have this coefficient of 16 in front of our X squared. We want to write this so that it's something squared in the denominator. So we have 16 in the numerator. How can we write it in the denominator. Well we can write this as X squared divided by 1/16, Dividing by 1/16 is the same as multiplying by 16. Recall your fraction rules and we have plus y squared. This is going to equal to one. Now we're almost there. But we want to write a denominator as something squared. How can we do that? Well right now we have X squared over 1/16. Now we can write 1/16 as 1/4 squared. So we have X squared over 1/4 squared plus Y squared. And I'm just going to write this is y squared over one square. Okay, that's equivalent just to make it clear what that denominator is and this is going to equal one. Now this isn't our standard form, we can pick out those important pieces of information which are gonna help us with our graph. So the x squared y squared terms, there's nothing being added or subtracted to the X and Y values before they're squared. And so H and cage are just zero. And since agent KR0, the sector is going to be just at the origin at 00 or at the origin. Now our value of A is 1/4. So our horizontal radius Is going to be 1/4. And vertical radius is going to be one vertical radius is a value and be in this case is one. Alright, we have our center, we have the horizontal and vertical radius. That's all we need in order to draw this. So let me give us some more space. Let's go ahead and draw it. So we have our axes here X M Y. More than a draw fairly large. And each of these is going to be one unit. All right. Now, what we wanna do is we want to start at the center, we're gonna start with the horizontal direction. So we're going to move in the horizontal direction. So that's going to be straight to the right and we're gonna move the distance of our radius which is 1/4. So we move 1/4. It's gonna be about here we draw a point And that point is 1/40. And we do the same thing in the negative direction. We start at the center, we move in the horizontal direction. In this case the negative horizontal direction, the distance of a rainy asse. 1/4. We draw a point and in this case this point is negative 1/40. Then we're going to do the same in the vertical direction. We go up a unit of one, our vertical radius from our center of 00. We get the .01 And then from our center we go down the vertical radius and we get the zero negative one. Now we can connect all these points and this is going to be our ellipse and it's going to look something like this. Great. We have our graph, that's part one of the questions. Now we need to figure out the focus and the focus are going to be two points whose sum of distances from any point on the ellipse is always the same. What does that mean? So we're going to choose or we're going to find two points in our lips if you take the distance from the first point to any point or to a point P on your lips and then you take the distance from the second point to that point p on your lips and you add those two distances together, you're going to get a value and that value is going to be the same no matter which point p you choose on the ellipse. Okay, you can choose any point along the entire lips in those distant. That sum of distances is going to be the same or folks are going to lie on the major axis. Okay. Which accesses that? Well in our case we have a vertical radius that is larger than the horizontal radius, which tells us that the vertical axis is going to be our major axis. Okay, so the vertical axis is going to be a major access. Our folks are going to lie along it, which means that our folks are going to be lying along this y axis. So what is the coordinates of the points of the folk? I well, we're going along this vertical axis. So the X value is going to be zero, just like the center of our ellipse. Then we're gonna start at the center and our folks are going to be an equal distance from our center. So they're going to be at plus or minus some value F along that y axis, That major axis. What's the value of F? Well, recall the F is given by the phone, F squared is going to be equal to b squared minus a squared. Now in this case we do b squared minus a squared because b squared is larger. If you have the case where a squared is larger, you would do these in the opposite direction, you would do a squared minus b squared. So in our case b squared is larger where we have a the vertical axis is our major axis. Okay, so we're doing it this way. Alright, so we get X squared is equal to one squared minus 1/ squared, which tells us that f squared is equal to one minus 1/16, which gives us an f squared value of 15/16. And I think the rating got cut off a little bit. So let's just move up so we can see all of that. So f squared is going to be 15/16 And if we keep working over here up here we're going to take the square root and we're gonna get that f is equal to plus or minus the square root of 15/16. Now we can simplify this, we know that the square root of 16 is four, so we can write the square root of 15/16 plus or minus the square root of 15 over four. Which tells us that our folks are going to be located at zero and plus or minus the square root of 15/4. So we have our graph and we have the location of our folk. I let's go ahead and see which answer choice corresponds to what we found. Now indeed, we see that the ellipse does not look like the ellipse. We drew the ellipse in in part D. Has a larger horizontal radius and a vertical radius. Okay, it's more stretched out along the sides and shorter talk to bow to. So that's not the right answer. In part C. Or choice C. The ellipse looks similar to the one we drew. It's long and skinny, has a longer vertical radius. It has the 10.10 point zero negative one and negative 0.250. Which were all the points we found along our graph. So the graph matches great. There, we just need to check the folk. I now the folk I N. C. R zero negative square root 15/4 and zero positive square root. 15/4. Which are the folk I we found. And so C is going to be the correct answer that matches what we drew and the focus 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. x² = 1 – 4y²

Key Concepts

Ellipses

Graphing Techniques

Foci of an Ellipse

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