In Exercises 1–18, graph each ellipse and locate the foci. x^2/25...

Question

In Exercises 1–18, graph each ellipse and locate the foci. x^2/25 +y^2/64 = 1

Accepted Answer

Hey, everyone. Welcome back in this problem. We're asked to graph the following ellipse and determine its F. And the equation we're given is X squared over 16 plus Y squared over 49 is equal to one. Now, when we're given an equation and we wanna graph it, we want to write it in the standard form of an ellipse. Now recall that the standard form is gonna be 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, if we write it like this, we can get important information about our lips from this equation. So in this equation, we know that the center is gonna be located uh H comma OK. We know that the horizontal radius is going to be given by A and we know that the vertical radius is going to be given by B. All right. So our equation we're given X squared over 16 plus Y squared over 49 is equal to one. Now, this is very close to being in standard form. The only thing we need to do is write these denominators as something squared. So X squared over 16, we can write 16 as four squared. And so we get X squared over four squared plus Y squared, over 49 we can write 49 as seven squared. And so we have Y squared over seven squared and all of this is equal to one. Now that we have it in this form, we're in standard form, we can pick out our center and our horizontal and vertical radius and draw our lips. So the center here, we have nothing being added or subtracted from our X and Y before they're squared. So that means that the value of H and the value of K are both zero. And so our center is going to be at 00 at the origin. That's the center of our lips or horizontal radiance, it's gonna be given by the A value. So the square root of the denominator associated with the X term, which is four in our vertical radius is going to be given by our B term, the square root of the denominator associated with the Y squared term, which is going to be seven. OK. With these three values, we can go ahead and graph our lips, we don't need to know anything else. So we have our X axis, we have our Y axis, we're just gonna draw out the scale. All right. So what we're gonna do is we're gonna start at the center which we know is at the origin 00, we're gonna go in the positive horizontal direction first and from the center, we're gonna move the distance of the horizontal radius which is four. So we're gonna go 1234 units over, we're gonna draw a point and that point is at four comma zero. And we're gonna do the same in the negative horizontal direction. We start at the origin our center and we move the horizontal radius which is four units, 1234, we draw a point and this point is going to be negative 40. Now we're gonna do the same in the vertical direction. We start at our center. We move the distance of the vertical radius up, draw point. This is the 0.07 and the same in the negative direction starting at the center moving down seven units when we get the 0.0 negative seven. Now that we have these four points, we connect them and these form our ellipse. So her lips is gonna look something like this. Let me fix this side here, there we go. So our lips look something like this. Our vertical radius is bigger than our horizontal radius. And so the vertical radius is going to be our major axis. And our folk when we're looking for our folk, they're gonna lie along the major axis. OK? So the folk are gonna be two points whose sum of distances from any point on the ellipse is always the same. So we're gonna find two points, the distance from the first point to a point P on the ellipse plus the distance from the second point to a point P on the ellipse is gonna give some value and that value is gonna be the same no matter which point P you choose along the ellipse and those are gonna be our folk. And again, our folk are gonna lie on our major axis, which we know in this case is the vertical axis because our vertical radius is larger than the horizontal radius. And because our ellipse is centered at the origin, our major axis is just going to be the Y axis. So our are gonna be given by the points while the X ray is going to be zero because we're along at Y axis, we're gonna start at the center and they're gonna be equally spaced from the center. So they're gonna be at zero plus or minus F in the Y direction for some distance F. Now, the question is, how do we find out? Well, recall that F is given by the following F squared is going to be equal to B squared minus A squared. We always take the larger first and then subtract the smaller. So in this case, B squared is larger than A squared. So we go B squared minus A squared. If it was the other way around, you would do a squared minus B squared. OK. So we have B squared minus A squared, F squared is therefore seven squared minus four square, which is equal to 49 minus 16, which gives us an F squared value of 33. And so F is going to be plus or minus the square root of 33. So our are gonna be located at zero and plus or minus the square root of 33. Now, if we wanted to draw these on our graph, we can do so. And we're gonna do that in ball blue. So our folk, we're going to be drawn in ball blue and we're at zero. OK. Again, along this Y axis and plus or minus route 33 that's gonna be under six and larger than five. So it's gonna be somewhere here in the positive direction and somewhere here in the negative direction, the location of our, all right. So if we look at our answer traces now that we've drawn our graph, we found our guy. We see that in answer D the ellipse doesn't look like the ellipse. We drew this one's wider, the horizontal radius is bigger than the vertical radius, which is not what we have. So we can cross that one out. If we look at c, the shape of the ellipse looks good. It looks similar to what we drew. We look at the points given we have 07, 400, negative seven and negative 40. Those are the points of our lips. And so this graph matches our lips. We just need to look at the folk and the folk are zero, negative route 33 and zero, positive route 33 which are the folk we found. And so C is going to be the correct answer that matches our graph and the FCA we found and that's it. 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. x^2/25 +y^2/64 = 1

Key Concepts

Ellipse Definition

Graphing Ellipses

Foci of an Ellipse

Watch next