In Exercises 1–18, graph each ellipse and locate the foci.4x²+16y² = 64

Question

Accepted Answer

Hey, everyone in this problem, we're asked to graph the following ellipse and determine its F. And the equation we're given is 25 X squared plus 81 Y squared is equal to 2025. Now, when we want a graph, an ellipse, the first thing we want to do is rewrite the equation so that it is in standard form. Now, we call it the standard form of the equation is X minus H squared over A squared plus Y minus K squared over B squared is equal to one. Now, when it's in this form, it's easier to graph because we can pick out certain pieces of information. And we know that H comma K is gonna be the coordinate of the center of our lips. We know that in this form A gives us the horizontal radius of our lips and we know that B gives us the vertical radius of our ellipse. Now, if you ever get stuck on mixing up A and B, which one relates to the horizontal radius, which one is the vertical radius, just take a look at the equation. We know that A squared is the denominator related to the X squared term. OK. X is our horizontal axis which tells us that A is going to be the horizontal radius. And similarly, B squared is the denominator for the Y squared term Y is the vertical axis. So B is going to represent the vertical radius. OK. So you can always look back at the equation to figure out which direction each of those relate to. Now, the equation we're given is 25 X squared plus 81 Y squared is equal to 2025. We have our X squared and Y squared term both on the, on one side which is numbers on the other. So we're close to our standard equation. But what we want to do is get the right hand side to be just equal to one instead of 2025. So in order to do that, we need to divide by 2025. So we get 25 X squared divided by 2025 plus 81 Y squared, divided by 2025 is equal to one. Now, we have our one on the right hand side and we just wanna write the coefficient in front of the X squared and Y squared terms as something squared in the denominator. So we can divide this left hand term 25 X squared divided by 2025 by 25. If we divide the numerator and denominator by the same thing. It's like we're just multiplying by one which doesn't change your equation. So we can write this as X squared over 2025 divided by 25. And similarly, for a Y term, we get Y squared divided by 2025/81. And we know that this is all equal to what if we simplify those denominators, we have X squared over 81 plus Y squared, over 25 is equal to one. And now we want to write these denominators as something squared. So we know that 81 is going to be nine squared. So we can write this as X squared over nine squared and we know that 25 is five squared. So we can write this as Y squared over five squared and this is all equal to one. Now that we have our equation written in this format, we can pick out the center, OK. We have H and K are both the zero in this equation. OK. We just have X squared and Y squared. We don't have anything added or subtracted. And so the center of our lips is going to be at 00 or at the origin. Now nine is gonna be our horizontal radius and the vertical radius is going to be five, right? And that's why we write it as something squared so that we can pick out that horizontal and vertical radius right from that equation. Um More simply. OK. So we have our center, we have our horizontal and vertical radius. That's everything we need in order to graph this ellipse. So let's go ahead and do it just gonna draw our axes. This is just gonna be a sketch. It's not gonna be perfect. That's OK. Let me just fix this. Vertical axis is a little bit curbed. There we go. All right, we're just gonna draw out our values. OK? So when we're drawing this, we're gonna start at our center, we know that our center is at 00. So we start at our center. Now, our horizontal radius is nine which means we're gonna move in the horizontal direction, OK? Straight to the right nine minutes, which is gonna give us this point here which is 90. We're gonna go back to our center and we're gonna do the same thing in the negative direction. So our horizontal radius is nine. So the distance from the center out to the left on a horizontal axis is also nine at this point here, which is gonna be negative 90. Now, we're gonna do the same with the vertical. We're gonna go back to the center and we're gonna go five units up and five units down in the vertical direction. So this 0.05 is part of our lips and same thing down here. The 0.0 negative five is part of our lips now we have this, these four points. So we just need to join them like in Ellipse. So it's gonna look something like this. Now, you can see that our horizontal radius is larger than our vertical radius radius. So the horizontal axis is going to be our major axis. All right. So we have our graph. We're also asked to determine the folk. Now the folk is going to be two points whose sum of distances from any point on the ellipse is always the same. OK? So what does that mean? So you're gonna choose two points in your lips. If you pick a point on your lips, say a point P, the distance from the first point to your point point P plus the distance from the second point to the point P. OK? That's gonna be your distance and that distance is the same for any point P you choose along the ellipse. OK? So you can choose any point along the ellipse and the distance from those two points is gonna be the same. Now, the folk, I always lie on the major axis and we know that the horizontal axis is our major axis in this case. So our folk are gonna be located at, well, our center is zero, they're gonna be on the major axis. So they're gonna be moving left or right. OK. So we're gonna have plus or minus F and the value of F we're gonna calculate in just a minute and then our Y axis is just gonna be zero because the center of our ellipses at zero and our folk are not going to be in that axis. So our folk are gonna be located at these two points. Let's find the value of F. Now, we can calculate F as F squared is equal to A squared minus B squared. Now we do it in the direction A squared minus B squared because our horizontal radius is larger. So A squared is larger than B squared. So we do it in this direction. If you had the case where B squared was larger, you would flip it, you would do B squared minus A squared. So we always do the larger one minus the smaller one. So this is gonna give us F squared is equal to nine squared minus five squared. So that gives us 81 minus 25. So we get that F squared and is equal to 56. This tells us that F is equal to plus or minus the square root of 56. We're just gonna try to simplify this a little bit. So we can write 56 as four times 14. We know how to take the square root of four. We know that it's a nice even number, nice um integer value. Our square root rules tells us that we can split these up. So we have the square root of four times the square root of 14. And so this is going to give us plus or minus two times the square root of 14. All right. So then our folk guy, let me draw this in blue are going to be located at plus or minus 2, 14 and zero. All right. So they're gonna be located somewhere about here in here. OK. On either side of the center, OK. Same distance from the center along our major axis. So that's it. We figured out our graph, we figured out our, let's go look at the answer traces. We see the answer choice D and answer choice B have the ellipse as like a longer and skinnier. OK. So it's the vertical radius is larger in these cases, which is not what we have. So we know we're looking at answer choice, either a or answer choice C now in C we have the points given as 0 25 81 00, negative 25 negative 81 0, which are not the points we found, right? We had the square roots. So we had 0590, et cetera. OK? So we know it's not C either. So let's, let's look at a, how does a look? Well, we have the points, we look, we found 050, negative 590, negative 90. And we have the FCA we found as well plus or minus two root 14 and zero. And so the answer here is going to be a, that matches what we found the love. 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.4x²+16y² = 64

Verified Solution

Key Concepts

Ellipse Standard Form

Foci of an Ellipse

Graphing Techniques