WEBVTT mathematics/algebra-2/eaton 00:00:00.000 --> 00:00:02.000 Welcome to Educator.com. 00:00:02.000 --> 00:00:08.500 Today we are going to talk about circles, beginning with the definition of a circle. 00:00:08.500 --> 00:00:17.900 A circle is defined as the set of points in the plane equidistant from a given point, called the center. 00:00:17.900 --> 00:00:31.500 For example, if you had a center of a circle here, and you measured any point's distance from the center, these would all be equal. 00:00:31.500 --> 00:00:44.600 And the radius is the segment with endpoints at the center and at a point on the circle. 00:00:44.600 --> 00:00:52.000 The equation for the circle is given as follows: if the center is at (h,k) and the radius is r, 00:00:52.000 --> 00:01:00.600 then the equation is (x - h)² + (y - k)² = r². 00:01:00.600 --> 00:01:07.000 And this is the standard form; and just as with parabolas, the standard form gives you a lot of useful information 00:01:07.000 --> 00:01:11.000 and allows you to graph what you are trying to graph. 00:01:11.000 --> 00:01:23.700 For example, if I were given (x - 4)² + (y - 5)² = 9, then I would have a lot of information. 00:01:23.700 --> 00:01:31.200 I would know that my center is at (h,k), so it is at (4,5). 00:01:31.200 --> 00:01:39.400 And the radius...r² = 9; therefore, r = √9, which is 3. 00:01:39.400 --> 00:01:48.000 So, based on this information, I could work on graphing out my circle. 00:01:48.000 --> 00:01:54.600 Use symmetry to graph a circle, as well as what you discover from looking at the equation in standard form. 00:01:54.600 --> 00:02:04.200 Looking at a different equation, (x - 1)² + (y - 3)² = 4: this equation describes the circle 00:02:04.200 --> 00:02:14.200 with the center at (h,k), which is (1,3); r² = 4; therefore, r = 2. 00:02:14.200 --> 00:02:19.400 So, I have a circle with a radius of 2 and the center at (1,3). 00:02:19.400 --> 00:02:36.300 So, if this is (1,3) up here, and I know that the radius is 2, I would have a point here; I would have a point up here. 00:02:36.300 --> 00:02:49.000 Symmetry: I know that, if I divide a circle up, I could divide it into four symmetrical quarters, for example. 00:02:49.000 --> 00:03:07.000 So, if I have this graphed, I could use symmetry to find the other three sections of this circle. 00:03:07.000 --> 00:03:18.000 All right, if the center is not at the origin, then we need to use completing the square to get the equation in standard form. 00:03:18.000 --> 00:03:29.700 Remember that standard form of a circle is (x - h)² + (y - k)² = r². 00:03:29.700 --> 00:03:33.100 With parabolas, we put those in standard form by completing the square. 00:03:33.100 --> 00:03:38.800 But at that time, we were just having to complete the square of either the x variable terms or the y variable terms. 00:03:38.800 --> 00:03:46.600 Now, we are going to be working with both; and as always, we need to remember to add the same thing to both sides to keep the equation balanced. 00:03:46.600 --> 00:04:00.900 If I was looking at something such as x² + y² - 4x - 8y - 5 = 0, 00:04:00.900 --> 00:04:09.500 what I am going to do is keep all the x variable and y variable terms together, and then just move the constant over to the right. 00:04:09.500 --> 00:04:18.400 The other thing I am going to do is group the x variable terms together: x² - 4x is grouped together, just like up here. 00:04:18.400 --> 00:04:26.200 And then, I am going to have y² - 8y grouped together, and add 5 to both sides. 00:04:26.200 --> 00:04:29.200 Now, I have to complete the square for both of these. 00:04:29.200 --> 00:04:36.800 This is going to give me x² - 4x, and then here I need to have b²/4. 00:04:36.800 --> 00:04:51.900 Since b is 4, that is going to give me 4²/4 = 16/4 = 4; so, I am going to put a 4 in here. 00:04:51.900 --> 00:05:07.200 For the y expression, I am going to have...let's do this up here...b²/4 = 8²/4, which is going to come out to 16. 00:05:07.200 --> 00:05:12.500 So, I am going to add 16 here; and I need to make sure I do the same thing on the right. 00:05:12.500 --> 00:05:21.200 So, I need to add 4, and I need to add 16; if I don't, this won't end up being balanced. 00:05:21.200 --> 00:05:35.400 Now, I want this in this form; so let's change it to (x - 2)² +...here it is going to be (y - 4)² =... 00:05:35.400 --> 00:05:41.500 4 and 16 is 20, plus 5; so that is 25. 00:05:41.500 --> 00:05:50.100 This gives me the equation in standard form; and the center is at (h,k), (2,4). 00:05:50.100 --> 00:05:55.500 And the radius...well, r² is 25; therefore, the radius equals 5. 00:05:55.500 --> 00:06:05.100 And as always, you need to be careful: let's say I ended up with something in this form, (x + 3)² + (y - 2) = 9. 00:06:05.100 --> 00:06:12.100 The temptation for the center might be just to put (3,2); but standard form says that this should be a negative. 00:06:12.100 --> 00:06:27.200 So, I may even want to rewrite this as (x - (-3))² + (y - 2) = 9, just to make it clear that the center is actually at (-3,2). 00:06:27.200 --> 00:06:30.400 And then, the radius is going to be the square root of 9, which is 3. 00:06:30.400 --> 00:06:35.800 So, be careful that you look at the signs; and if the signs aren't exactly the same as standard form, 00:06:35.800 --> 00:06:45.600 you need to compensate for that, or even just write it out--because -(-3) would give me +3, so these two are interchangeable. 00:06:45.600 --> 00:06:53.400 All right, in this example, we are asked to find the equation of the circle which has a diameter with the endpoints (-3,-7) and (9,-1). 00:06:53.400 --> 00:06:57.700 So, let's see what we are working with. 00:06:57.700 --> 00:07:13.200 Just sketch this out at (-3,-7), right about there; over here is (9,-1); there we have the diameter of the circle. 00:07:13.200 --> 00:07:19.600 We have a circle like this, and we want to find the equation. 00:07:19.600 --> 00:07:29.000 Recall that the formula for the equation of a circle is (x - h)² + (y - k)² = r². 00:07:29.000 --> 00:07:32.800 Therefore, I need to find h; I need to find k; and I need to find the radius. 00:07:32.800 --> 00:07:39.800 Recall that (h,k) gives you the center of the circle. 00:07:39.800 --> 00:07:47.300 Since this is the diameter of the circle, the center of this line segment is going to be the center of the circle. 00:07:47.300 --> 00:07:57.200 So, (-3,-7)...over here I have (9,-1); and here I have the center--the center is going to be equal to the midpoint of this segment. 00:07:57.200 --> 00:08:10.900 Recall the midpoint formula equals (x₁ + x₂)/2, and then (y₁ + y₂)/2. 00:08:10.900 --> 00:08:26.800 So, the center--the coordinates for that are going to be equal to (-3 + 9)/2, and then (-7 + -1)/2, 00:08:26.800 --> 00:08:40.200 which is going to be equal to 6/2...-7 and -1 is -8/2; which is equal to (3,-4). 00:08:40.200 --> 00:08:46.100 This means that h and k are 3 and -4; so I have h and k; I need to find the radius. 00:08:46.100 --> 00:08:54.900 Well, half of the diameter...this is all the diameter; this is my midpoint; and I know that this is at (3,-4). 00:08:54.900 --> 00:08:58.800 So, I just need to find this length--this is the radius. 00:08:58.800 --> 00:09:08.800 And I now have endpoints; so I can use either one of these--I have a set of endpoints here and here, and here and here. 00:09:08.800 --> 00:09:13.500 I am going to go ahead and use these two, and put them into the distance formula. 00:09:13.500 --> 00:09:21.700 (-3,-7) and (3,-4)--I can use these in the distance formula: the distance here equals the radius, 00:09:21.700 --> 00:09:34.200 which is the square root of...I am going to make this (x₁,y₁), and then this (x₂,y₂). 00:09:34.200 --> 00:09:49.200 So, this is going to give me...x₂ is 3, minus -3, squared, plus...y₂ is -4, minus -7, squared. 00:09:49.200 --> 00:09:59.600 So, the radius equals the square root of 3 + 3; a negative and a negative is a positive; all of this squared, 00:09:59.600 --> 00:10:07.700 plus -4...and a negative and a negative is a positive, so -4 + 7, squared. 00:10:07.700 --> 00:10:19.200 So, the radius equals the square root of...3 + 3 gives me 6, squared, plus...7 - 4 gives me 3, squared. 00:10:19.200 --> 00:10:31.700 So, the radius equals √(36 + 9); 36 plus 9 is 45, so the radius equals √45. 00:10:31.700 --> 00:10:51.400 But what I really want for this is r², so r² is going to equal (√45)², which is going to equal 45. 00:10:51.400 --> 00:10:58.500 Putting this all together, I can write my equation, because I now have h; I have k; and I have r. 00:10:58.500 --> 00:11:11.800 So, writing the equation up here gives me (x - 3)² + (y...I can either write this as - -4, 00:11:11.800 --> 00:11:19.400 or I can rewrite this as (y + 4)² = r², which is 45. 00:11:19.400 --> 00:11:29.300 So, this, or a little more neatly, like this: (y + 4)² = 45--this is the equation for the circle. 00:11:29.300 --> 00:11:33.500 And I found that information based on simply knowing the diameter. 00:11:33.500 --> 00:11:38.200 Knowing the diameter, I could use the midpoint formula to find the center, which gave me h and k. 00:11:38.200 --> 00:11:47.100 And then, I could use the distance formula to find the distance from the center to the end of the diameter, which gave me the radius. 00:11:47.100 --> 00:11:51.300 And then, I squared the radius and applied it to that formula. 00:11:51.300 --> 00:11:56.000 Example 2: Find the center and radius of the circle with this equation. 00:11:56.000 --> 00:11:59.500 In order to achieve that, we need to put this equation in standard form. 00:11:59.500 --> 00:12:10.100 And recall that standard form of a circle is (x - h)² + (y - k)² = r². 00:12:10.100 --> 00:12:21.800 So, we need to complete the square: group the x variable terms together on the left; also group the y variable terms on the left. 00:12:21.800 --> 00:12:26.600 Add 8 to both sides to move the constant over. 00:12:26.600 --> 00:12:39.200 I need x² - 8x + something to complete the square, and y² + 10y + something to complete the square, equals 8. 00:12:39.200 --> 00:12:52.700 So, for the x variable terms, I want b²/4, and this is going to be 8²/4, or 64/4, equals 16. 00:12:52.700 --> 00:12:54.200 So, I am going to add 16 here. 00:12:54.200 --> 00:13:05.000 For the y variable terms, b²/4 is going to equal 10²/4, which is 100/4, which is 25. 00:13:05.000 --> 00:13:24.800 I need to be careful that I add the same thing to both sides to keep this equation balanced, so I am also going to add 16 and 10 to the right side. 00:13:24.800 --> 00:13:29.000 Correction: it is 16 and 25--there we go. 00:13:29.000 --> 00:13:39.800 (x - 4)² equals this perfect square trinomial, and I am trying to get it in this form; that is what I want it to look like. 00:13:39.800 --> 00:13:50.600 Plus...(y + 5)² comes out to this perfect square trinomial. 00:13:50.600 --> 00:14:00.200 On the right, if I add 8 and 16 and 25, I am going to end up with 49. 00:14:00.200 --> 00:14:07.700 8 and 16 is going to give me 24, plus 25 is going to give me 49. 00:14:07.700 --> 00:14:20.800 Now, I have this in standard form: because the center of a circle is (h,k), I know I have h here, 00:14:20.800 --> 00:14:29.100 and I know I have k here, this is going to give me...h is 4; k is -5. 00:14:29.100 --> 00:14:40.500 Be careful with the sign here, because notice: this is (y + 5), but standard form is - 5, so this is equal to (y - -5)²--the same thing. 00:14:40.500 --> 00:14:44.600 It is just simpler to write it like this; but make sure you are careful with that. 00:14:44.600 --> 00:14:52.400 The radius here: well, I have r²; the radius, r², I know, is equal to 49. 00:14:52.400 --> 00:14:58.300 Therefore, r = √49, so the radius equals 7. 00:14:58.300 --> 00:15:09.200 Therefore, the center of this circle is at (4,-5), and the radius is equal to 7. 00:15:09.200 --> 00:15:19.600 Example 3: Find the radius of the circle with the center at (-3,-4) and tangent to the y-axis. 00:15:19.600 --> 00:15:24.500 This one takes more drawing and just thinking, versus calculating. 00:15:24.500 --> 00:15:33.600 The center is at (-3,-4), right about here. 00:15:33.600 --> 00:15:39.000 The other thing I know about this circle is that it is tangent to the y-axis. 00:15:39.000 --> 00:15:48.200 That means that, if I drew the circle, it is going to extend around, and it is going to touch this y-axis. 00:15:48.200 --> 00:15:54.100 Well, the radius is going to have one endpoint on the circle, and the other endpoint at the center. 00:15:54.100 --> 00:16:05.400 Therefore, the radius has to extend from (-3,-4) over here. 00:16:05.400 --> 00:16:16.000 And at this point, we are able to then find the length, because, since x is -3, and it has to go all the way to x = 0, 00:16:16.000 --> 00:16:24.200 then this distance must be 3; therefore, the radius equals 3. 00:16:24.200 --> 00:16:28.000 And again, that is because I know the center is here at -3, and I know 00:16:28.000 --> 00:16:36.300 that the other endpoint of the radius is going to be out here, forming the circle, 00:16:36.300 --> 00:16:42.700 and that, because it is tangent to the y-axis, x is going to be equal to 0 right here. 00:16:42.700 --> 00:16:53.100 So, I know that x is equal to 0 here; and I know that x is equal to -3 over here; so it is just 1, 2, 3 over--this distance here is going to be 3. 00:16:53.100 --> 00:16:58.000 And that is going to be the same as the radius, so the radius is equal to 3. 00:16:58.000 --> 00:17:05.700 Example 4: Find the equation of the circle that is tangent to the x-axis, to x = 7, and to x = -5. 00:17:05.700 --> 00:17:20.300 We are given a bunch of information about this circle and told to put it in standard form. 00:17:20.300 --> 00:17:29.100 The first thing we are told is that it is tangent to the x-axis; so this circle is touching the x-axis; let's just draw a line here to emphasize that. 00:17:29.100 --> 00:17:40.900 It is also tangent to x = 7; x = 7 is going to be right here--it is tangent to this. 00:17:40.900 --> 00:17:48.300 It is also tangent to x = -5, out here. 00:17:48.300 --> 00:17:53.800 I am going to end up with a circle that is touching, that is tangent to, these three things. 00:17:53.800 --> 00:17:56.700 Let's think about what that tells me. 00:17:56.700 --> 00:18:01.600 I need to find h and k (I need to find the center). 00:18:01.600 --> 00:18:06.200 I also need to find the radius, so I can find r². 00:18:06.200 --> 00:18:17.200 If this extends from -5 to 7, that gives me the diameter. 00:18:17.200 --> 00:18:24.400 So, the diameter goes from 7 all the way to -5; so if I just add 7 and 5 (the distance from here to here, 00:18:24.400 --> 00:18:30.300 plus the distance from here to here), I am going to get that the diameter equals 12. 00:18:30.300 --> 00:18:37.000 The radius is 1/2 the diameter, so the radius equals 6. 00:18:37.000 --> 00:18:41.700 I found that the radius equals 6. 00:18:41.700 --> 00:18:47.100 The radius is going to extend from these endpoints to the center. 00:18:47.100 --> 00:18:55.700 And I know that it is 6, so I know that the radius is going to go from 7 over here, 6 away from that. 00:18:55.700 --> 00:19:01.200 7 - 6 is 1; it is going to go up to x = 1. 00:19:01.200 --> 00:19:12.500 Again, that is because the length of the radius is 6, so the distance between the center and this endpoint has to be 6. 00:19:12.500 --> 00:19:16.200 7 - 6 is 1; the radius is going to extend from there to there. 00:19:16.200 --> 00:19:23.700 Therefore, the x-value of the center has to be 1. 00:19:23.700 --> 00:19:31.000 Now, what is the y-value of the radius? The other thing I know is that this circle is tangent to this x-axis. 00:19:31.000 --> 00:19:35.100 So, I know that it is going to have an endpoint on the x-axis. 00:19:35.100 --> 00:19:53.300 And then, if it is going to extend from here to here, it is going to have to go up to 6; therefore, the center is at (1,6). 00:19:53.300 --> 00:19:59.700 All right, so the radius equals 6, and the center is at (1,6); and that gives me an equation: 00:19:59.700 --> 00:20:08.300 (x - 1)² + (y - 6)² = the radius squared. 00:20:08.300 --> 00:20:12.500 If r = 6, then r² = 36. 00:20:12.500 --> 00:20:21.400 Again, that is based on knowing that this is tangent to x = -5, x = 7, and the x-axis. 00:20:21.400 --> 00:20:31.000 So, I had the diameter, 12; I divided that by 2 to get the radius; I know that I have an endpoint here and an endpoint at the center. 00:20:31.000 --> 00:20:33.100 So, that gives me the x-value of the center, which is 1. 00:20:33.100 --> 00:20:38.500 I know I have an endpoint here, and also an endpoint at the center; so it has to be up at 6. 00:20:38.500 --> 00:20:43.500 That gives me (1,6) for my value. 00:20:43.500 --> 00:20:47.500 OK, and this is just drawn schematically, because the center would actually be higher up here. 00:20:47.500 --> 00:20:57.700 This is just to give you...the center is actually going to be up here, now that I have my value: it is going to be at (1,6). 00:20:57.700 --> 00:21:03.000 OK, that concludes this lesson of Educator.com on circles; thanks for visiting!