Pre Calculus 8.2 - Circles and Ellipses

Summary

In this Pre-Calculus session, led by John Swanson, we delve into the captivating world of circles and ellipses. While circles are beautifully simple, they're merely special cases of ellipses, which are defined by the constant sum of distances from any point on the ellipse to the two fixed foci. This lesson navigates us through equations that define these shapes, their properties, and intriguing real-world applications, like planetary orbits and medical technologies. With a mix of technical details and quirky facts, Swanson brings these mathematical concepts to life, ensuring that learners can graph and understand the subtleties of these conic sections with confidence and curiosity.

Highlights

• Explore how circles are just special types of ellipses! 🔵✨
• Learn the equations that define these elegant curves, and how to manipulate them! 📏✏️
• Discover real-world applications of ellipses, from orbits to acoustics! 🌎🔊
• John Swanson simplifies complex concepts with engaging explanations! 📚🎙️

Key Takeaways

• Ellipses are cooler than you think—they're just circles with a twist! 🤯
• The sum of distances from any point on the ellipse to two foci is constant! 🎯
• Ellipses have everyday applications like in 'whisper rooms' and kidney stone treatments! 🚀
• Understanding the eccentricity of ellipses tells you how circle-like they are! 🔵➡️🟢
• Graphing ellipses is like connecting dots, only more elliptical! 🎨

Overview

In this engaging and informative session, John Swanson walks us through the enchanting world of circles and ellipses, two fundamental components of Pre-Calculus. His detailed explanations unravel how circles are merely special cases of ellipses, defined by all points equidistant from a center, contrasting beautifully with ellipses' two-foci definition.

Swanson skillfully uses real-life applications to illustrate these concepts, explaining how ellipses are pivotal in architectural designs like whisper rooms, where sound clarity is uniquely perfected, and in medical procedures like breaking kidney stones using focused sound waves. The planetary orbits icing on the cake shows just how vast the applications stretch.

Through vivid examples and calculations, Swanson ensures students not only understand but also appreciate these geometric wonders. His clear, methodical approach demystifies graphing and understanding the properties of ellipses, equipping students with the tools to solve complex problems with confidence.

Chapters

• 00:00 - 01:30: Introduction to Circles and Ellipses In the chapter 'Introduction to Circles and Ellipses,' the focus will be on understanding ellipses, with an emphasis on their similarities and differences to circles, which are considered a special case of ellipses. Key learning objectives include analyzing equations of ellipses, graphing ellipses, finding equations of ellipses that meet specific conditions, solving application problems involving ellipses, and solving systems of non-linear equations.
• 01:31 - 02:30: Concept of Circle in Conic Sections A circle is defined as all points equidistant from a center point.
• 02:31 - 05:30: Understanding Ellipses This chapter explores the geometric concept of ellipses, starting with the basic properties and equations of a circle. It explains the formula for a circle as x squared plus y squared equals r squared, which is derived from the familiar distance formula. The discussion also considers scenarios when the center of the circle is not at the origin (0, 0) but instead at any point (h, k). In such cases, the equation adapts to (x - h)² + (y - k)² = r². This understanding forms the foundation for comprehending the more complex structure of ellipses, where the axes may vary in length.
• 05:31 - 08:30: Applications of Ellipses The chapter discusses the mathematical concept of ellipses, focusing on their geometric definition. It begins with a recap of the distance formula relevant to circles and contrasts it with how ellipses are defined. Specifically, an ellipse is described as the set of points where the sum of the distances to two fixed points, known as foci, remains constant. This definition is central to understanding the properties and applications of ellipses in various fields.
• 08:31 - 14:30: Equations of Horizontal and Vertical Ellipses The chapter titled 'Equations of Horizontal and Vertical Ellipses' explores the geometric properties of ellipses, emphasizing the concept of foci. It details that any point on the ellipse maintains a constant sum of distances to the two foci. This characteristic feature defines the relationship between points on the ellipse and its foci, illustrating the foundational principle that m (distance to one focus) plus n (distance to the other focus) equals a constant.
• 14:31 - 16:30: Graphing Ellipses The chapter titled 'Graphing Ellipses' introduces the concept of ellipses, emphasizing their significance similar to parabolas. It highlights the importance of the foci in ellipses, comparing it to how parabolas are structured and mentioning that unlike parabolas, ellipses are closed forms.
• 16:31 - 24:30: Conclusion and Summary The chapter discusses the reflective property of ellipses, where any signal originating from one focus will be reflected to the other focus. This reflection characteristic is a key principle in understanding acoustics and how sound behaves in elliptical structures. Additionally, there is a mention of 'whisper rooms', which are specific spaces designed with acoustics in mind, leveraging the properties of ellipses. These whisper rooms are rare and exist in a few locations in the world.

Pre Calculus 8.2 - Circles and Ellipses Transcription

• 00:00 - 00:30 welcome back to chapter eight in this section section two we will be looking at circles and ellipses now most of what we'll be looking at will be ellipses because circles are really just a special case of an ellipse and we'll see the similarities and differences there are learning targets we'll analyze the equation of an ellipse graph ellipses find equations of ellipses that satisfy the given conditions solve application problems involving ellipses and then solve systems of non-linear
• 00:30 - 01:00 equations uh we actually just did that in the last chapter too so first question what is a circle remember all of these conic sections have definitions that involve distance a circle is all points equidistant from a center point so we have that center point the center of the circle and then all the points that are the same distance which is the radius
• 01:00 - 01:30 away from that center point and so the equation of a circle is x squared plus y squared equals r squared that's the distance formula remember distance would be the square root of the x distance squared plus the y distance squared if we weren't centered at 0 0 we're at h k we have x minus h and y minus k each of them squared equals r squared
• 01:30 - 02:00 and here's where that distance really gets in if you square it that you have x minus the x 1 squared plus y minus y 1 squared the first one was at 0 zero so the minus zeros kind of go away so that was a circle what's an ellipse an ellipse is all points where the sum of the distances from the point to two fixed points foci that would be the plural of focus
• 02:00 - 02:30 right you have one focus you have two foci just like you have one mongoose you have two mongai is constant there's that constant sum there so here we have up any point on the ellipse the distance from this one plus the distance to that one is going to be equal to some constant sum m plus n equals a constant
• 02:30 - 03:00 to so that's that's what an ellipse is since we have the ellipse here uh some of the um some of the applications of ellipses um kind of like with parabolas those focuses the foci are important with a parabola though it wasn't closed in here with an ellipse
• 03:00 - 03:30 anything coming from this focus going out to the ellipse is going to reflect to the other focus and same with the other direction as well so we have there it goes there we can and there it all reflects from one focus gets reflected to the other focus um we have there are some things called whisper rooms uh there's a few of them in the world
• 03:30 - 04:00 those are ellipsoid rooms so it's a three-dimensional ellipse well half of one because it's going to be a flat ground a three-dimensional ellipse would be kind of like egg-shaped um but it's half one if you stand at one focus and you whisper someone at the other focus can hear you even though no one else can because it takes all of the sound and it reflects it right back into that
• 04:00 - 04:30 one point um the using um the high frequency sounds or low frequency sounds or whatever to get rid of kidney stones ultra high frequency shock waves to get rid of kidney stones is the same thing because if you have someone sitting on one focus and you have the little sound emitter on the other focus it's going to hit those kidney stones
• 04:30 - 05:00 from every single direction with those things and it breaks them up um other applications of ellipses planetary orbits are elliptical they're fairly circular most of them but they are elliptical the sun is at one of the foci what's it the other one i don't know but it's different for each planet
• 05:00 - 05:30 so that's just some of the um applications of these ellipses so let's look at formula type stuff we have horizontal ellipses and we have vertical ellipses the horizontal ones are wider than they are tall the vertical ones are taller than they are wide i know that the way i have my variables is
• 05:30 - 06:00 a little bit different than the book so and i'll explain the differences there as we get there so we have foci the foci are along the long axis the long axis is called the major axis first of the center is where it's centered
• 06:00 - 06:30 the foci are along the like here we have them here and here on the long axis so horizontally or vertically the vertices are going to be the points on the ellipse that are at the end of that axis so we can see them
• 06:30 - 07:00 here so they're either horizontal or vertical the other ones are called co-vertices oh there we have i forgot my order here the major axis connects the two vertices so it's the long one it goes through the vertices the focus and the center
• 07:00 - 07:30 so there it is there it is um the co-vertices are the other vertices right so if we have the horizontal ones then they'd be the vertical so they're the ones here here here and here are the co-vertices
• 07:30 - 08:00 the minor axis is the line that connects the co vertices and then we also actually have the semi-major axis and the semi-minor axes um those don't go all the way across they go from the center out to a vertex or the center to a co-vertex so it's half
• 08:00 - 08:30 there are there we go so some equations if the center is at 0 0 which the 2 that i have graphed are at 0 0. if the center's at 0
• 08:30 - 09:00 0 the equation is going to be x squared over a squared plus y squared over b squared equals 1. where a is that the horizontal distance and from the center to the the vertex or co-vertex and the b is the vertical distance from the center to the vertex or co-vertex so a is your horizontal radius b is your vertical radius now they're not technically radii but
• 09:00 - 09:30 that's basically kind of what they are um now the book does this differently the book makes the a the long one always and b is always the short one it makes it so a couple things don't have to change but a lot of things do i like having a being the horizontal one b being the vertical one it doesn't really make much of a difference
• 09:30 - 10:00 so when the center is 0 0 we just have x squared over a squared plus y squared over b squared equals 1. now with a circle that horizontal and that vertical is the same and so a squared and b squared are the same thing and so we multiply them over and that's where we get x squared plus y squared equals r squared so that's why i say a circle is just a special case of a para of an ellipse it's where the a squared and the b squared are the same
• 10:00 - 10:30 in fact in a circle the two foci then you get closer and closer and closer together it becomes more and more circular until they're both on the center and that's a circle that's why we don't have foci in a circle it's because they're both on the center if the center's at h comma k we get x minus h y minus k plug those in x minus h squared over a squared plus y minus k squared over b squared
• 10:30 - 11:00 equals one notice these all equal one we always want we divide suit equals one that's where we get the a squared and the b squared and then a squared minus b squared equals c squared c is that distance from the center to the focus that's how we can find c a squared minus b squared equals c squared it's not part
• 11:00 - 11:30 of the equation itself um but it's there now this is one thing where having the a being the big one and b being the small one um makes it work a little bit better where me having a being horizontal and b being vertical is a little bit different because in a vertical one a squared minus b squared is negative well absolute value and we solve that problem
• 11:30 - 12:00 so it's the big one minus the small one so the center of our ellipses are either going to be 0 0 or h k the vertices so the vertices are going to be a away horizontally and then that's in the x direction so it'd either be plus or minus a comma
• 12:00 - 12:30 0 or h plus or minus a comma k the co-vertices are going to be in the y direction they're the b's so it's either 0 comma plus or minus b or h comma k plus or minus b we're just adding that value to the center the foci are going to be the same as the vertices only plus or minus c so we have plus or
• 12:30 - 13:00 minus c comma 0 and h plus minus c comma k for vertical ones the center it's either 0 0 or h k for the vertices we're going to go how i have it it's going to be that b distance because it's vertical if you look at the ones in the book they use a
• 13:00 - 13:30 but either way it's vertical so it'll be 0 plus or minus b there we go zero plus or minus b or h comma k plus or minus b so the co vertices and the vertices just switch places horizontal to vertical and that's true for the other way around
• 13:30 - 14:00 too these ones are going to come down here so we'll have plus or minus a comma zero or h plus or minus a comma k the foci it doesn't go straight over because in the horizontal it was horizontal and the vertical has to be vertical but it'll be the same as the vertices only we use c so zero plus minus c or h comma k plus or minus c um we have another thing that we deal with with ellipses
• 14:00 - 14:30 and that's called the eccentricity which actually does come from the word eccentric which means a little bit off and that's actually what an ellipse is it's how far off are they how much unlike a circle are they so if the centricity is zero it's actually a circle the further away from zero it gets
• 14:30 - 15:00 the less like a circle it looks like and the more oblong it is and what eccentricity is how you find it is it is just the um it's the focus length c divided by the semi major axis so the long side so c over a
• 15:00 - 15:30 or in a vertical one it could be c over b again the bigger it is the more um the the bigger it is the the more stretched out it is the smaller it is the more to a circle it is until c equals zero it's a circle um the closer c gets to a or b
• 15:30 - 16:00 the longer it gets and so this will always be between zero and one because these the focus will never be outside of the ellipse so what are we going to do with these things we could find the constant sum this is distance formula so if we have two foci and a point on the ellipse the distance from the focus to one point plus the distance from the other focus to the point will have to be equal to the same thing at any given point that's the constant
• 16:00 - 16:30 sum so we can set it up in distance formula we have one point five comma zero and then one focus five comma zero to zero twelve so negative five minus zero squared plus zero minus negative twelve squared plus the other one five minus zero squared plus 0 minus negative 12 squared gives us 25 plus 144 square root
• 16:30 - 17:00 plus another 25 plus 144 square root so we get root 169 plus root 169 which is 13 plus 13 which is 26 and this will actually be the length of the major axis as well so like horizontally this would be 2a vertically this is 2b that's the constant sum it's the length of that major axis that long axis
• 17:00 - 17:30 we could be writing equations in standard form so if we have a center at zero zero and a vertex at negative nine zero um and we need another one a covert x at zero seven we have the a and the b the vertex is horizontal so that means it'll be under the x's we have 9 squared under the y's we'll
• 17:30 - 18:00 have 7 squared 9 squared is 81. 7 squared is 49 so it's just x squared over 1 plus y squared over 49 equals 1. because again that's because it's at 0 0 and we had the two values that go in there now if it's 0 0 we have a covert x at 10 0 so notice this is horizontal
• 18:00 - 18:30 so that means under x still is going to be 10 squared and a focus is 0 24. we need to find that other value so we know that we'll have x squared over 10 squared plus y squared over b squared equals one now here's where that is so the big one minus a small one since this is the co-vertex
• 18:30 - 19:00 this is a small one so we're going to have b squared minus 10 squared equals 24 squared we can add it over to get b squared equals 676. and a lot of times people find b at this point they're like okay so we need to square root to get something that doesn't come out right you don't need to square root because
• 19:00 - 19:30 we have b squared in the equation just plug in 676. so we get x squared over 100 plus y squared over 676 equals 1. now if you do not realize that that's a co-vertex so that's b squared minus that um that becomes problematic because you'd actually end up with another number that a lot of times actually it kind of makes
• 19:30 - 20:00 sense um it doesn't work so you have to make sure you have the order there right let's say we have a center at negative four seven vertex at negative four negative three focus at negative four zero so what we can see from here center to the vertex what's changing is that y value which means this is a vertical one
• 20:00 - 20:30 and that that major axis going from negative 3 to 7 or from 7 to negative 3 either way is going to be 10 so we'll have a y squared over 10 but we also need the y minus 7 squared the x plus 4 squared and we know that this number right down here is going to be that 10 value
• 20:30 - 21:00 c is going to be 7 because that's the difference between that and that so we can find the equation we have 10 squared minus a squared equals 7 squared again it's 10 squared because that one's first that one's that one's the big one that's the vertex length we can square them out we can subtract
• 21:00 - 21:30 to get negative a squared equals negative 51 so a squared equals 51. plug that in plug in the 10 squared to get that equation graphing graphing an ellipse so we have x squared plus y squared or x squared over 36 plus y squared over 81 equals one so if we're graphing the ellipse we start out with the center and then we have our horizontal radius
• 21:30 - 22:00 so it's the square root of that so six one two three four five six both directions we have our vertical radiuses square root of 81 so 9. and then we're going to connect these in an elliptical fashion and then we'll see how close
• 22:00 - 22:30 my drawing is now again as we can tell i am not an artist i don't expect you to be artists but notice this is not a diamond i did not connect these things with straight lines do try to make it round how close was i
• 22:30 - 23:00 i was off a little bit but you know it wasn't wasn't i've done worse and drawing on my computer is a little bit tougher so what if it's not centered at zero zero it's okay do the same thing so we're centered at five comma negative two horizontal radius is square root of 16 is four
• 23:00 - 23:30 so we go each direction vertical radius square root of 36 is six and then connect
• 23:30 - 24:00 all right that one also actually looked pretty decent especially that top half i kind of nailed the top half of that one so um drawing ellipses graphing ellipses start with your center do your vertical radar your horizontal radius your vertical radius connect the dots so these are circles and ellipses
• 24:00 - 24:30 again mostly ellipses because the circle is just that special case um yeah so we're working on this time so i will see you in class but until then keep working problems keep having questions and as always happy mathing