In Exercises 1–18, graph each ellipse and locate the foci. 7x² = ...

Question

In Exercises 1–18, graph each ellipse and locate the foci. 7x² = 35-5y²

Accepted Answer

Everyone in this problem. Whereas to graph the following ellipse and determine its foca. And the equation we're given is 10 Y squared equals minus three X squared. Now when we're given an equation of the lips, were asked to graph it, the first thing you want to do is write it in standard form. Okay, so the standard form of an ellipse recall is x minus h squared over a squared plus. Why minus K squared over B squared is equal to one. Now when we have our lips in this standard equation we know that the center of our lips is that the coordinate H comma K. And we know that our horizontal radius is given by A and our vertical radius is given by B. So let's start with our equation 10 Y squared equals 30 minus three X squared. And we want to write it in this form that we've written here. So what we wanna do is we want to have the X and the Y terms on one side and we want to get one on the other side. So let's start by moving this negative three X squared to the left hand side. We're gonna have three X squared plus 10 Y squared is equal to 30. And then on the right hand side we want to have a constant of just one instead of 30. So we need to divide everything by 30. So we get three X squared plus 10, Y squared divided by 30 Is equal to one. Now when we have 3/ This is going to give us a coefficient of 1/10. So we get x squared over 10. And then for our y squared term we have a coefficient of 10/30 that's equivalent to one third. So we get y squared over three is equal to one. Now when we look at our standard equation we also see that in the denominator we have a squared and B squared. So we want these terms in the denominator to be squares of something. So in order to write these as squares. Well what is 10 going to be equivalent to 10 is going to be equivalent to the square root of 10 squared? And similarly three is going to be equivalent to the square root of three squared. So now we have our equation in that form that we wanted and we can compare. Okay so are H and K are just zero. We have just X squared and just sorry y squared. Not like you to hey x squared and y squared. We don't have any subtraction or addition on those terms. And so our center, the center of our lips is going to be at the origin. It's gonna be at 00. Alright and then if we're looking at are horizontal and vertical radius, let me move down and write this out as we go. So we know that our center is that the origin? Our horizontal radius Is going to be the squared of 10. Okay that's always the horizontal radius is always the denominator. That value that squared okay with the extra. So it's gonna be the square root of 10. And the vertical radius is a similarly that denominator the square root kind of this this term that's squared on the y squared term. And so that's gonna be the square root of three. Alright, so we have our center, we have our vertical radius and we have our horizontal radius so we can go ahead and draw this out to finish that first part of the problem and then we can work on our focus after. So we have our X and y axes 123 or. Alright, so the center of our lips is at the origin. So we're centered right here at the origin. Our horizontal radius is the square root of 10. Okay, so starting at our center which is at the origin, we're going to go in the horizontal direction. Square root of 10 units. Now we know the square root of nine is three. So the square root of 10 is just gonna be a bit over three. So 123 in a little bit. So this is going to be the squared of 10. So we have a point here and we also know from the center we're going to go in the negative X direction. The negative horizontal direction, that same amount of units is a squared of 10. We have negative square root of 10 here. All right, so we have these two points in on our X axis. Now we're gonna do the same for the Y. So we started our center and in the vertical direction, in the y direction we go up square root of three units. Somewhere around here. And we do the same thing from the center, in the negative direction, the square root of three units. Somewhere around here. So we have those four points now and we can connect them. We connect these, oops, hopefully with a smoother line than that. There we go. Alright, let's just label these points. So we don't forget what they are. So this point up top they were at an X value of zero and we went up to a Y value of square root of three. And similarly down here it's a value of negative. The square root of three on the right hand side, this is an X value of the squared of 10, A Y value of zero. And on the other side squared of negative square root of and zero. Okay, so those are the four points that we use to make our lips. Now we have. So the problem asks us to graph it and it also asked us to determine its focus. So we've done the graph let's move on to the folk. I now we're talking about folk. I what we mean here is two points whose sum of distances from any point on the ellipse is always the same. Okay, so you can choose two points in our lips. Okay, If you take the distance from those points to a point P on the ellipse. Okay, so you take the distance from the first point to a point P. You add it to the distance from the second point to point P. That sum is going to be the same for any point P. You choose around the ellipse. Okay, those are are folks that we're looking for now, those are going to lie on the major axis which accesses major. Well, let's look the access with the larger radius is going to be our major axis. So in this case the horizontal radius is larger. So our x axis or our horizontal axis is going to be our major axis. So we know that our focus are going to lie along the x axis. Alright, So because they're on the X axis, they're going to be an equal distance from the center and so they're going to be located at our center which is zero plus or minus some value F. That we're going to calculate and the y value of our center is just zero. So they're going to be located at this point here or these points, how do we calculate alphas F is calculated as F squared is equal to a squared minus b squared. Now I write a squared minus B squared because the value of a is larger than the value of B. If the value of b squared was larger we would actually put this. So we always take the larger of a squared B squared and subtract the other. So in this case we have that F squared is equal to the square root of 10 squared minus the square root of three squared. We're just gonna continue up here. So F squared Is equal to seven. 10 squared 10 squared is 10 squared of three squared is 3. 10 minus three gives us seven. And so our value of F is going to be plus or minus the square root of seven. And so our Foca are going to be located at zero plus or minus a squared of seven and zero. So if we were to draw these on our graph, they're gonna be located somewhere about here. We know that the square root of seven is going to be bigger than two because seven is bigger than four and it's going to be smaller than three because seven is less than nine. And we know that the square root of nine is three. Okay, so they're gonna lie somewhere here and those are going to be our Foca. Alright, so if we look at our answer choices starting with D. We know we can eliminate d because it has a larger vertical radius than horizontal radius. Okay, it's kind of stretched up instead of stretched out. So we know we can eliminate that one. What about sea? The sea looks good. We have all of these points that we've found. Zero. Um Plus a squared of 30 minus the square root of three squared of 10. 0 negative the square root of 10 0. Okay. And the folk I hear are at plus or minus the square root of +70. Which is exactly what we found. Let's just make sure that nothing else again, B is stretched upwards instead of outwards. And in a The points are at 10 and zero and 0 and three not the square roots. And so c. is the correct answer. The folk I. R. At negative the square root of seven, positive the square root of seven. And the graph is the looks like the graph that we drew on the side there. 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. 7x² = 35-5y²

Key Concepts

Ellipse Definition

Conic Sections

Foci of an Ellipse

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