In Exercises 1–18, graph each ellipse and locate the foci. x^2/(9...

Question

In Exercises 1–18, graph each ellipse and locate the foci. x^2/(9/4) +y^2/(25/4) = 1

Accepted Answer

Hey everyone welcome back in this problem. We are asked to graph the following ellipse and determine its Boakye. And the equation we're given is x squared divided by 25/4 plus Y squared divided by 49/4 is equal to one. The first thing we want to do when we're given in ellipse and were asked to graphic is to rewrite the equation in standard form. Now we call it, the standard form of the ellipse is x minus h squared over a squared plus Y minus K squared over B squared is equal to one. We want to write it like this because the standard form gives us important information about the ellipse. So if we have our equation in this form, we know that the center of our lips is going to be located at the coordinates H comma K. We know that the horizontal radius is going to be given by A and that the vertical radius is going to be given by B. So the equation were given were divided by 25/4 plus. Why squared divided by 49/4 is equal to one. This is really close to being in standard form, but what we want to do is write the denominators as something squared. Okay, we have a squared and B squared. So we want to write these as something squared. Now, for the X squared we have 25 divided by four. We can write this as 5/2 squared because five squared is 25 2 squared is four. And for the y squared term we have 49/4. We can write this as 7/2 squared, seven squared is 49 2 squared is four. So we get 7/2 squared and all of this is equal to one. Now that we're in standard form, we can pick out those important pieces of information. So the X. And the Y terms don't have anything added or subtracted before they're squared. That means that the H and K. Values are zero. And so the center it's just going to be at the origin 00. Now our horizontal radius Is going to be our a value which is 5/2 and our vertical radius is going to be our B. Value and the square root of the denominator associated with the Y. Term. It's gonna be seven halves. So we have our center our horizontal and vertical radius and that's enough to draw this ellipse. So if we draw our axes here we have our X axis under y axis. We're just going to kind of mark out the values so that we can grab this. All right. So we're gonna start at the center of our ellipse, which we know is at the origin. We're gonna go in the horizontal direction first. So we're gonna go in the positive horizontal direction. So straight to the right and we're gonna move a distance of our horizontal radius, which is five halves or 2/5. Once we get there we put a point and that point is going to be on on our lips. And in this case that point is 5/2. Then we do the same thing in the negative horizontal direction. We start at our center, we move our horizontal radius in the negative horizontal direction. We put a point. That point is -5/2. And we have those. Now we're gonna move to the vertical direction. Do the same thing, we start at the center, we move up our horizontal radio or horizontal direction At a value of 7/2. We at a point that point is 07 halves and that's going to be on our lips. And then we do the same in the negative direction. We have the 70.0 negative 7/2. And we combine all of these things to draw our lips. Now this drawing isn't great. The spacing of all of these ticks isn't perfect because we've drawn it by hand. But what I want you to notice is that our vertical radius is larger than the horizontal radius. And so this distance top to bottom is actually larger so it's more spread out top to bottom and narrower side to side. Okay, Alright. So we've got our graph and I've just said that it's taller and skinnier. And so if we look at our answer choices, we see that answer A and answer C. Have similar shaped ellipsis. The ellipse in B and D. Are more squished top to bottom and wider. So we know that we can already eliminate those. So let's work on our focus right now. And the folks are gonna be two points whose sum of distances from any point on our lips is always the same. So we have two points in our lips. If you take the distance from one of those points to a point on the ellipse P You added to the distance from the 2nd point to that Point P. And you get a value and that value is going to be the same no matter which point P on the ellipse you choose any point along that ellipse that you choose P. That value is going to be the same. The folks are going to lie along the major axis and in our case the vertical radius is larger, which tells us that the vertical axis is going to be our major axis. So if these lie along our vertical axis, that's going to be along the war axis. So the X value for these points is going to be zero. Okay, it's just going to stay at our center. The y value is going to start at zero. Okay. And these are going to be an equal distance from the center. So these are going to be located at plus or minus some distance F from the center. How do we find F. Well, we can calculate F through the following F squared is going to be equal to b squared minus a squared. Now we do b squared minus a squared. Because our vertical radius is larger, B squared is larger than a squared. Okay, if it was the other way around, you would switch these. So F squared is going to be equal to b squared. Are vertical radius squared 7/2 squared minus a squared. Or horizontal radius squared five squared. This is gonna give us F squared is equal to 49/4 minus 25/4, Which is equal to 24/4, which gives us a value of six. And so if we take the square root, we're going to get the F Is equal to plus or minus the square root of six. And so our focus are going to be located at zero plus or minus square root of six. Those two points there. Okay, so we've got our graph, we found the location of our focus. We can go ahead and pick an answer. We already eliminated D. By saying that the ellipse was not shaped like our lips. The vertical axis in our lips is larger than the horizontal. And that's not the case in D. In C. The shape of the ellipse looks correct. However, the points are 07500, negative seven negative 50. Which are not the points that we found B. We already eliminated based off the shape just like D. In A. We have the 0.3 point 52.500, negative 3.5 and negative 2.50. We have the folk I at zero and negative route six and zero and positive route six. So that is going to be the correct answer. We have A. We have those folks I we found. And we have the graph that matches the graph that we drew. Thanks everyone for watching. I hope this video helped you win the next one.

In Exercises 1–18, graph each ellipse and locate the foci. x^2/(9/4) +y^2/(25/4) = 1

Key Concepts

Ellipse Definition

Graphing Ellipses

Foci of an Ellipse

Watch next