In Exercises 1–18, graph each ellipse and locate the foci. 25x²+4y² = 100

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Hey everyone in this problem, we are asked to graph the following the lips and determine its folk. I in the equation we're given is 196 X squared plus Y squared is equal to 3136. Now when we're given an ellipse and we want to graphic, we want to rewrite the equation were given in standard form and recall it. The standard form of an ellipse is given by x minus h squared over a squared plus y minus K squared over B squared is equal to one. Now we want to write it like this because we're able to pick out important pieces of information about our lips from the equation. In this form, the center of the ellipse in this given this equation is going to be at the coordinate h comma K. The horizontal radius is going to be given by A and the vertical radius is going to be given by B. So the equation were given 196 X squared plus 16 Y squared is equal to 3136. Now, all of the x squared y squared terms are on one side with number on the other side. So it's close However, we want that number on the right hand side to be one, not 3136. So we need to do is divide both sides of our equation by 3136. We're going to have 196 x squared divided by 3136 plus 16, Y squared divided by 3136 is equal to one. And now we just need to simplify these coefficients on the left hand side. In our standard equation we want no coefficient in front of the X. But we wanted to nominate her of something squared. So we have 196 divided by 3,136. Okay, they're both divisible by 196. So if we divide the numerator and the denominator by 196, our equation is going to be equivalent because we're essentially multiplying by one And we're gonna get x squared over 16. We're gonna do the same thing with the y squared term. We have 16 over 3136. They can both be divided by 16. So we divide the numerator and denominator by 16 and we get y squared over 196. And all of this is equal to one. Now we have one step left to do, we have these denominators but we want to write them as something squared. So in order to write them as something squared, let's take a look, we have X squared over 16. We know we can write 16 as four squared. So we get X squared over four squared and then we have y squared over 196. We know that we can write 196 as 14 squared. So we get y squared over 14 squared is equal to one. Now our equation is in that standard form and we can pick up those important pieces of information that are going to help us draw our ellipse, stop the center of our lips. Well, our agent K values are just zero. We have nothing added or subtracted from our X and Y values before they're being squared. So our H and K values are zero. And so our center is just going to be at the origin 00. Now, in terms of our radius, the horizontal radius Is going to be our a value, which is four horizontal radius of four and a vertical radius of 14. That's our B value. So with our center, our horizontal radius and our vertical radius, we can go ahead and draw this graph. Now our vertical radius is larger than our horizontal radius, which tells us that our major axis is the vertical axis. So that's just an important piece of information. We just keep that in the back of our mind as we're working through this. And that's gonna come into play when we start talking about the folk. I no to draw this though, we have our Y axis and X axis. We're just going to label And each tick is going to be two units. All right now we're gonna start at our center, which we know is the origin. and in the horizontal direction, we're gonna go a distance of our horizontal radius, in the positive horizontal direction. Okay, so to the right, we're moving four units. Each of these ticks is going to be too, so we're going to move four units and we are going to have a point here, we're gonna do the same in the negative horizontal direction. K are horizontal radius is four. So in the negative horizontal direction, we move four units, which takes us to this point here, which is negative for zero. The positive was positive for zero. Now we're going to do the same in the vertical direction. So we started the origin in the positive vertical direction. We moved 14 units are vertical radius. That's gonna take us to a point here at zero and 14. And we did the same in the negative vertical direction. We started our center, we moved that vertical radius down. We get a point at 0 -14. Now that we have these four points, we can connect them. Hmm, what looks like in the lips. Something shaped like in the lips. We're gonna get something that looks kind of like this, ours is a little wonky just by the hand drawing. But we have now an idea of the shape of our ellipse. Alright, so we've done our graph. Now we need to move on to our focus. What is a focus, what what are the folk? The folk are going to be two points whose sum of distances from any point on our lips is always the same. Okay, so we get two points in our lips. We take the distance from one point to a particular point P on our lips and then added to the distance from the second point to that point P. We're going to get some value and that value is going to be the same no matter which point P. We choose along the ellipse. So those are folk. I and the folks are going to lie on the major access. Now our major axis is the vertical axis. This means they're going to lie along this y axis. So we know that the X coordinate is going to be zero with our center is at zero and we're going in the vertical direction and then they're going to be at our center zero plus or minus some value F in the Y direction. And we're going to calculate what that value F is going to be and that's gonna be the location of those two foca. So let's give ourselves some more room here. No F squared is going to be given by b squared minus a squared. Now we take b squared minus a squared because b b squared is larger. If a squared is larger, you would reverse the direction, you end up a squared minus b squared. In that case you would have your major access lying in the horizontal direction. In this case we have major access being the vertical access, B squared is larger, so we have b squared minus a squared. So this is going to give us f squared is equal to 14 squared minus four squared. F Squared is equal to 180. And if we continue up over here we take the square root and we get the f is equal to positive or negative square root of 180. Now we want to try to simplify the square root of 180 as much as we can. So let's think of factors of 180 that have a nice square root. Okay, that are perfect squares. Well, we can write 180 as times five using our square square root rules. Sorry, we can write this as a square root of 36 Times The Square Root of five. Okay, we can break up those two values and we get an F value Of plus or -6 times a squared of five because the square root of 36 is six. And so our f value is plus or minus six times the square root of five, which tells us that our folks are going to be located at zero And plus or -6. Route five. Okay, so those are our folk, I now looking at our answer choices, let's start with D because it's the one right here. Now, the shape of this ellipse looks good. It looks similar to what we've drawn, it's long and skinny, but the points don't match. These points are at 0 81 16, 00, negative 81 negative 16 0. So that is not the correct answer. If we look at options, see the ellipse is kind of going in the wrong direction. We have the bigger horizontal radius and a smaller vertical radius. And so this is kind of stretched it widthwise, whereas ours is stretched top to bottom. So C. Is not the correct answer either. If we look at B, the shape looks good, it's taller and skinnier. We have the 0.0 14 400, negative 14 and negative 40, which are the points we had on our lips. So the graph matches perfectly. And now let's look at the sky. The folks out here are zero, negative six, route five and zero positive six. Route five. Which are the folk I we found. And so B is the correct answer here and matches the work that we showed below. 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. 25x²+4y² = 100

Verified Solution

Key Concepts

Ellipse Standard Form

Foci of an Ellipse

Graphing Techniques