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 +y^2/36= 1

Accepted Answer

Hey everyone in this problem, we are asked to graph the following lips and determine its folk. I and the equation we're given is x squared over four plus Y squared over 64 is equal to one. Now, when we're given an ellipse and we want to graph it. What we want to do is write it in the standard form of the equation. And the standard form for an ellipse is given by x minus h squared over a squared plus y minus K squared over B squared is equal to one. And recall when you have this standard equation of an ellipse, we can get some important pieces of information about our lips from the equation. In this form, we know that our sector is going to be located H comma K in the center of our lips. 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 for our equation we were given x squared over four, course Y squared over 64 is equal to one. This is very close to being in standard form, however, we want the denominators to be something squared. So we have x squared over four. We know that we can write four as two squared. So we can write this as x squared over two and then we have plus y squared over 64. We know that we can write 64 as eight squared. So we can write this as y squared over eight squared and all of this is equal to one and I'm sorry, I missed the square on the two there. So it should be X squared over two squared so that it's equivalent to X squared over four. Alright, So now this is in standard form and we can compare to the general equation in order to find the center and the vertical and horizontal radius. So the X and Y terms have nothing added or subtracted from them before they're squared. That means that the agent K values are just zero, which tells us that the center of our lips Is that 00 at the origin. Now our horizontal radius this is given by the a value or the square root of the denominator on the X square term which is gonna be too and our vertical radius is given by our B. Value which is the square root of the denominator associated with the y squared term which is eight. So with those three pieces of information we can draw this out. So let's draw our diagram. I'm just gonna label the axis Y axis, X axis and then the negative axes. Alright so How we're going to draw this, we're gonna start at the center which we know is at 00. Now our horizontal radius is two. That means the distance from the center to the points on the X axis ours too. So we started the center, we're going to go in the positive horizontal direction. So along the positive X axis and we're going to move a distance of 212, we're gonna draw a point there And this point is located at 20. Then we're going to do the same in the negative direction. We start at the center, we moved to in the negative extraction 12. We have a point there, we label it -20. Now we're going to do the same in the vertical. So we started our center, our vertical radius is eight. So from the center in the positive y direction, we're going to move eight units 12345678. We draw a point. This is the 0.0 comma eight. And we did the same in the negative direction from the center. Eight units down. We have a point and this is the 0 -8. And we want to connect all of these points with a line and this is gonna be our lips. Okay, so our lips is going to look something like this. We have those four points and this is a general shape. Now we've got our graph, we want to find the focus. The folk. I are going to be two points whose sum of distances from any point on the ellipse is always the same. What does that mean? We're going to 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 going to give us some value and that value is going to be the same no matter which point P. We choose along the lips. Now our folk, I always lie on our major axis. What is our major access in this case? Well the vertical radius is larger than the horizontal radius. And so our major axis is going to be the vertical axis. So our folks are going to lie on the vertical axis in this case our center is at the origin. So our vertical axis is just the y axis. And so our points are folk, I are going to lie on the y axis. So we know that they're going to have an X value of zero and they're also going to start with the y value of zero and we're going to add something. Okay, now the folk are going to be an equal distance from the center and so we're going to add or subtract some distance F. So our points are going to be zero and plus or minus some distance F. That are folk I. R from the center. No. How do we find f Well recall. But if squared is going to be equal to b squared minus a squared because our vertical axis is a major axis, we take the vertical radius squared minus the horizontal radius squared. If it was the other way around we would do a squared minus b squared. We always want to take the bigger one first. So we have f squared is equal to b squared minus a squared. So f squared is equal to eight squared minus two squared. This is going to be equal to 64 -4. And so we get an f squared value continuing up here of 60, You can take the square root to get that f is going to be positive or negative square root of 60. Now we want to try to simplify this. So what we want to do is look for factors of 60 that are perfect squares. We can write 60 as four times 15 and we know that four is a perfect square. We can break up this route so we have the square root of four times the square root of 15. Okay, recall your rules for dealing with square roots And this gives us plus or -2 times route 15 since the square root of four is 2. Alright, so now we can write Our focus as zero And plus or -2 or 15, those are the coordinates of the two of them and they're going to be an equal distance from that center. So we've done the graph, we found our focus. Now we just need to choose an answer choice. So if we look at answer choice, D we see that the shape of the ellipse is not like the shape we drew. This ellipse is very short and very wide. Our lips was tall and narrow so this does not match the shape C. C. Does match the shape. Okay, the general shape looks good. But if you look at the points in C, we have 0 64 400, negative 64 negative 40. Those don't match the points that we found to be on our ellipse. So it's not see be just like d doesn't match the shape of the ellipse. We drew a The shape looks good. If we look at the points, we have 08 to 00 negative eight and negative 20 all points that we're on our lips. And the focal here are located at zero negative to route 15 and zero positive to route 15. Which are the folks that we found. And so the answer is going to be a this matches the graph and the focus that we found. Thanks everyone for watching. I hope this video helped you in the next one.

In Exercises 1–18, graph each ellipse and locate the foci. x^2/9 +y^2/36= 1

Key Concepts

Ellipse Definition

Graphing Ellipses

Foci of an Ellipse

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