WEBVTT mathematics/geometry/pyo
00:00:00.000 --> 00:00:02.300
Welcome back to Educator.com.
00:00:02.300 --> 00:00:09.500
For this lesson, we are going to go over equations of circles.
00:00:09.500 --> 00:00:20.100
The standard equation of a circle when there is a center (h,k) (and that is the coordinates of the center)
00:00:20.100 --> 00:00:33.600
and a radius of r units is going to be (x - h) (this is the x-coordinate of the center), squared, plus (y -...
00:00:33.600 --> 00:00:39.300
and this is the y-coordinate)...squared, equals r², the radius squared.
00:00:39.300 --> 00:00:47.500
Now, be careful: the x-coordinate is going to go with the x, and the y-coordinate is going to go with the y.
00:00:47.500 --> 00:01:00.800
And don't forget: this number here is the radius squared; it is not the radius itself; it just has to be that number, squared.
00:01:00.800 --> 00:01:17.000
Example: If I give you the center as (1,1) and the radius is 2, then to write an equation, it is going to be (x -...
00:01:17.000 --> 00:01:27.600
the x-coordinate is 1), squared, plus (y - 1)², equals...not 2; be careful with that;
00:01:27.600 --> 00:01:37.100
it is 2 squared, so that is going to be 4; and this would be the equation of that circle.
00:01:37.100 --> 00:01:51.400
Now, if I give you an equation, (x - 2)² + y² = 16 (that is the equation of a circle),
00:01:51.400 --> 00:02:01.800
and I ask you for the center and the radius, the center, (h,k)...the x-coordinate is going to be the number that is grouped with the x;
00:02:01.800 --> 00:02:10.100
that is 2; be careful--this is not -2, because the standard is going to be x minus the number.
00:02:10.100 --> 00:02:17.500
So, if there is a minus there, then it is just the 2 that is going to be the x-coordinate.
00:02:17.500 --> 00:02:28.800
Keep in mind, also: if this is a negative number, it is going to change to a plus; so instead of x - 2, it would be x + 2, if this was a -2.
00:02:28.800 --> 00:02:39.500
And then, for the y-coordinate, there is no number that is grouped with the y to become a binomial.
00:02:39.500 --> 00:02:49.900
So, this right here is going to be 0, because this is like saying (y - 0)².
00:02:49.900 --> 00:03:06.900
So then, my y-coordinate is 0, and my radius is the square root of 16, which is 4.
00:03:06.900 --> 00:03:16.400
Given the equation, you can solve for your center, or you can find your center and the radius.
00:03:16.400 --> 00:03:24.900
And also, if you are given the equation of a circle, and you are asked to graph, then you would first plot the center,
00:03:24.900 --> 00:03:33.800
which is (2,0); (2,0) is right there--that would be your center; and then, the radius is 4.
00:03:33.800 --> 00:03:41.300
Now, we cannot count diagonally on the coordinate plane, so we can only count up, down, left, and right.
00:03:41.300 --> 00:03:55.900
So, I can count 1, 2, 3, 4, up 4; I can count down 4: 1, 2, 3, 4, because then, from the center, that would make 4 units; the radius is 4.
00:03:55.900 --> 00:04:03.800
1, 2, 3, 4: the same thing works with that; 1, 2, 3, 4.
00:04:03.800 --> 00:04:09.900
And then, what you are going to do is connect the dots, not like a square, but you know that it is going to be a circle--
00:04:09.900 --> 00:04:25.400
it is the equation of a circle--so it is going to be something like that.
00:04:25.400 --> 00:04:36.800
Again, after finding the center, (h,k), the radius would be this number's square root; that will be 2.
00:04:36.800 --> 00:04:40.800
If you are given the equation, that is how you would find the center and the radius.
00:04:40.800 --> 00:04:47.600
And if you are given the center and the radius, then you would have to make sure that the x-coordinate is paired up with the x,
00:04:47.600 --> 00:04:57.000
the y-coordinate with the y, and then square the radius to give you the constant and then the equation.
00:04:57.000 --> 00:05:02.800
Our examples: Determine the coordinates of the center and the radius for each equation.
00:05:02.800 --> 00:05:08.600
The first one: there is the equation; I want to find the center and the radius.
00:05:08.600 --> 00:05:23.000
The center would be...3 is my x-coordinate, because that is the one that is paired with the x; 3, comma, and 7.
00:05:23.000 --> 00:05:30.600
If it has a minus, that means that the coordinate is a positive number.
00:05:30.600 --> 00:05:40.500
And the radius is going to be the square root of that number; so the square root of 9 is 3.
00:05:40.500 --> 00:05:53.300
For this one, the center is going to be 3/4...and for this one, it says y + 5/8.
00:05:53.300 --> 00:06:03.700
If it is a plus, if it changed to a plus, that means that this number is a negative number; so that is negative 5/8.
00:06:03.700 --> 00:06:18.800
And then, the radius is the square root of 4, which is 2.
00:06:18.800 --> 00:06:35.300
OK, and then, the third one: here, this is in the correct form...plus y²...and then, the constant itself should be on the other side.
00:06:35.300 --> 00:06:41.200
This is where the constant should go, on the right side by itself.
00:06:41.200 --> 00:06:49.200
Since it is not there, I need to move it there; so I am going to add the 100; that way,
00:06:49.200 --> 00:07:06.000
my equation is going to be (x - 10)² + y² = 100; and that would be standard form of a circle.
00:07:06.000 --> 00:07:18.000
Now that it is in standard form, my center will be 10, comma...and don't forget that this right here
00:07:18.000 --> 00:07:33.000
is the same thing as (y - 0)²; so my y-coordinate is 0, and then the radius is √100, so that is 10.
00:07:33.000 --> 00:07:42.100
So again, make sure that your equation is written in standard form before you find your center and your radius.
00:07:42.100 --> 00:07:53.200
Otherwise, your radius would be 0; so be careful with that, just to make sure; you have to add the constant over to this side.
00:07:53.200 --> 00:08:00.900
For the next example, we are going to write an equation for each of these, based on what information is given to us.
00:08:00.900 --> 00:08:06.100
The first one: they give us the center, and they give us the radius.
00:08:06.100 --> 00:08:15.000
In that case, we can just go ahead and plug each of these in: this is (h,k), and this is r.
00:08:15.000 --> 00:08:33.200
Remember that the standard form of the equation of a circle is (x - h)² + (y - k)² = the radius squared.
00:08:33.200 --> 00:08:44.300
The equation is going to be (x - -3)...minus a negative...that means I have a plus 3, squared,
00:08:44.300 --> 00:08:53.000
plus (y + 2)² (it is a negative number), equals...and then for the radius squared,
00:08:53.000 --> 00:09:01.700
the constant, don't forget to square these numbers: this is going to be 25/4.
00:09:01.700 --> 00:09:04.900
And that is the equation of that circle.
00:09:04.900 --> 00:09:23.100
The next one is the same thing: (x - 0)² + (y - 1/2)² = √8, squared.
00:09:23.100 --> 00:09:34.500
Here, you want to simplify this; if it is x - 0, then that is the same thing as x², plus (y - 1/2)²,
00:09:34.500 --> 00:09:41.000
equals...the square root of 8, squared, is just 8; remember that the square root and the square cancel each other out.
00:09:41.000 --> 00:09:44.900
So, that is the equation of that circle.
00:09:44.900 --> 00:09:53.500
And then, for the next one, they give us the center, and instead of giving us the radius, they give us a point on the circle.
00:09:53.500 --> 00:10:10.600
The point on the circle is (4,0); now, in order for me to find the equation of a circle, I need the center and the radius.
00:10:10.600 --> 00:10:17.600
I need these two things; instead, I have the center, and I have a point on the circle.
00:10:17.600 --> 00:10:22.100
I have this; now, I have to find the radius...so how do you find the radius?
00:10:22.100 --> 00:10:29.800
If you have the center, and you have a point on the circle, then you just have to find the distance between those two.
00:10:29.800 --> 00:10:40.400
So then, if they just give you the center, and they give you a point on the circle, if you find the distance, then that will give us the radius.
00:10:40.400 --> 00:10:55.100
So, you can use the distance formula; that is going to be (x₂ - x₁)² + (y₂ - y...
00:10:55.100 --> 00:11:05.700
oh, I said y, and I wrote x...y₁)².
00:11:05.700 --> 00:11:19.700
That will be x₂...that is 4, minus the 4, the two x's...the difference of them is 0, squared is 0.
00:11:19.700 --> 00:11:32.300
Plus...0 minus -5; that is 5; squared is 25; so distance equals √25, which is 5.
00:11:32.300 --> 00:11:39.300
That means that the radius is 5; the center is that; the radius is 5.
00:11:39.300 --> 00:11:52.000
Now, I can go ahead and write my equation: (x - 4)² + (y + 5)²
00:11:52.000 --> 00:12:00.300
(and that is a plus because it is a negative) = 5²; that is 25.
00:12:00.300 --> 00:12:05.200
And that is it; everything is simplified; that is the answer.
00:12:05.200 --> 00:12:12.900
And then, this last one, endpoints of a diameter: instead of giving us the center, and then a point on the circle,
00:12:12.900 --> 00:12:18.800
they are giving us the two points on the circle that make up a diameter.
00:12:18.800 --> 00:12:26.900
So then, it is like giving us this and this; they are giving us these two endpoints.
00:12:26.900 --> 00:12:31.900
So, how do you find the center? Now again, we still need the center, and we still need the radius.
00:12:31.900 --> 00:12:41.200
We have to look for both of those things; so if they give us the endpoints of the diameter, to find the center,
00:12:41.200 --> 00:12:48.500
we have to find the midpoint of those two points; the midpoint of a diameter is the center.
00:12:48.500 --> 00:12:58.600
How do you find the midpoint? You use the midpoint formula, which is x₂ + x₁,
00:12:58.600 --> 00:13:08.900
or x₁ + x₂, divided by 2, and y₂ + y₁, divided by 2.
00:13:08.900 --> 00:13:13.300
You are just going to add up the x's and divide it by 2, add up the y's and divide it by 2.
00:13:13.300 --> 00:13:24.000
It is also like average; you only have two x's, so you add them up and divide by 2; that is the average of the x's, comma, the average of the y's.
00:13:24.000 --> 00:13:44.600
The midpoint of this is going to be 1 + 5, over 2, comma, the y's: -1 + 3, over 2; that is 6/2, which is 3, and this is 2/2, which is 1.
00:13:44.600 --> 00:13:54.700
So, this is the midpoint; that means that this is the center; we found the midpoint from this point to this point: that is (3,1).
00:13:54.700 --> 00:14:01.700
The center is (3,1); now, I have to find the radius.
00:14:01.700 --> 00:14:07.400
So, just like we did for this problem here, if you have the center, and you have a point on the circle
00:14:07.400 --> 00:14:14.700
(which is both of those), then you can just find the distance, and then find the radius, like that.
00:14:14.700 --> 00:14:21.900
You can also find the distance between these two (find the distance of this to this) and divide it by 2
00:14:21.900 --> 00:14:29.300
(diameter divided by 2 is radius), or you can just take the center (since we found the center, (3,1)),
00:14:29.300 --> 00:14:38.900
and use one of these to find the distance, because from the center to this point is a radius; or the center to this point is a radius.
00:14:38.900 --> 00:14:52.800
So, you can just pick whichever one looks easier; this one looks easier to me, so I am going to use this point and the center, and find the distance between them.
00:14:52.800 --> 00:15:00.000
The distance is going to be...I am going to use this one, and I am going to use this one;
00:15:00.000 --> 00:15:25.100
so (1 - 3)² + (-1 - 1)² is going to be the square root of...2² is 4, plus (-2)² is 4.
00:15:25.100 --> 00:15:36.600
The distance equals the square root of 8, or it can be 2√2.
00:15:36.600 --> 00:15:45.300
That is the distance between the center and then one of the points; that makes it the radius: 2√2,
00:15:45.300 --> 00:15:53.800
or it can also be √8, because I know that I am going to have to square it to give me the equation.
00:15:53.800 --> 00:16:02.400
My equation, then, is going to be x minus the x-coordinate of the center; that is that; squared--don't forget the squared;
00:16:02.400 --> 00:16:21.000
plus y minus 1, squared; equals the square root of 8, squared (is 8); that is the equation of that circle.
00:16:21.000 --> 00:16:26.500
So again, if you are given the endpoints of a diameter, you have to find the midpoint to get the center.
00:16:26.500 --> 00:16:34.100
And then, you have to use the distance formula to find the distance between the center, and then one of these points, to find the radius.
00:16:34.100 --> 00:16:39.300
Just, whatever they give you, always make sure that you have the center and the radius.
00:16:39.300 --> 00:16:43.500
If they don't give you either one of these, then you have to look for it; but you need the center,
00:16:43.500 --> 00:16:49.700
and you need the radius, in order for you to write the equation of the circle.
00:16:49.700 --> 00:16:54.900
The third example: we are going to graph each of the circles.
00:16:54.900 --> 00:17:12.100
So again, I need center, and I need radius; the center, this one, is 0, the x-coordinate; this one is 2; and the radius is 2.
00:17:12.100 --> 00:17:26.100
Graph the center first: (0,2) is right there, and then the radius is going to be up 2, down 2, right 2, left 2.
00:17:26.100 --> 00:17:34.500
And then, you are going to just draw a circle.
00:17:34.500 --> 00:17:49.500
The next one: here, our center is (1,-1); the radius is √20, which makes that 2√5.
00:17:49.500 --> 00:17:58.900
Now, since we are going to have to count this on the coordinate plane, it would just be best to find √20 in decimals.
00:17:58.900 --> 00:18:19.000
So, you can just use your calculator; I have a calculator on my screen; so √20...you should get about 4.47, which is around 4.5, 4 and 1/2.
00:18:19.000 --> 00:18:51.200
So, the center is (1,-1), and then I am going to go up 4 and 1/2: 1, 2, 3, 4, 4 and 1/2; 1, 2, 3, 4, 4 and 1/2; 1, 2, 3, 4; and then, 1, 2, 3, 4.
00:18:51.200 --> 00:19:18.600
Again, draw your circle, something like that.
00:19:18.600 --> 00:19:31.000
The next one: we are given the graphs of the circles, and then we have to find the equation.
00:19:31.000 --> 00:19:39.300
The diameter is from here to here; I have a diameter from here to here.
00:19:39.300 --> 00:19:44.400
That means that the center (the midpoint of the diameter) is going to be right there.
00:19:44.400 --> 00:19:50.900
Now, you can count it; so then, this is 5 here, and this is -5 here; that makes the diameter 10.
00:19:50.900 --> 00:20:05.000
I have 10; that is 5; the radius is going to be 5; so this is r, which is 5; so that means it should be 5 each way.
00:20:05.000 --> 00:20:27.000
The center is (0,0), and the radius is 5; then, the equation becomes (x - 0)² + (y - 0)² = radius squared, 25.
00:20:27.000 --> 00:20:42.600
So, we have to simplify this; we don't want x - 0, so it is going to be x² + y² = 25, and that is your equation.
00:20:42.600 --> 00:20:57.900
For this one, look for your diameter: there, there, there, and there; so this is -1, and this is +7;
00:20:57.900 --> 00:21:19.700
so that means that the distance from my diameter is 8; half of that is...let's see, this is (7,1), and this is (-1,1);
00:21:19.700 --> 00:21:26.000
so then, it is 8; that means that half of that is 4; my radius has to be 4;
00:21:26.000 --> 00:21:36.400
that means it has to go 4 this way, 4 that way, and each of the ways.
00:21:36.400 --> 00:21:47.600
And then, that means that my center is at (3,1), and then the radius is 4.
00:21:47.600 --> 00:22:02.600
Once you have the center and the radius, you can write your equation: (x - 3)² + (y - 1)² = 16.
00:22:02.600 --> 00:22:09.100
Is that simplified? Yes, so that is the equation of the circle.
00:22:09.100 --> 00:22:14.300
OK, we are going to do one more example; we are going to have a fifth example for this one.
00:22:14.300 --> 00:22:22.500
This one is a little bit challenging, but we are going to just go ahead and try it.
00:22:22.500 --> 00:22:38.000
The graphs of x = 1 and y = -2 are tangents to the circle with the center in the second quadrant and a diameter of 8.
00:22:38.000 --> 00:22:42.100
Write the equation of the circle.
00:22:42.100 --> 00:22:54.000
They give us two equations: these are linear equations--that means two lines are tangent to the circle.
00:22:54.000 --> 00:23:02.500
We know that tangents are lines that touch the circle (intersect the circle) at one point.
00:23:02.500 --> 00:23:26.000
So then, let's draw these lines here: x is 1--that is this side right here; y = -2 is right there, so this is x = 1; this is y = -2.
00:23:26.000 --> 00:23:35.500
And they are tangents; that means that the circle is going to be touching this line and this line one time.
00:23:35.500 --> 00:23:42.700
The center is in the second quadrant; that means that somewhere in here is going to be the center.
00:23:42.700 --> 00:23:53.800
The diameter is 8; so if it is going to be touching this line and this line with the center in the second quadrant,
00:23:53.800 --> 00:24:02.800
that means that it is going to look something like this.
00:24:02.800 --> 00:24:16.600
If the diameter is 8, then the radius is going to be 4; so tangent to the center is going to be 4 (4: distance away from this line).
00:24:16.600 --> 00:24:24.600
So, 1, 2, 3, 4; it is going to be somewhere here; the center is going to be somewhere there.
00:24:24.600 --> 00:24:32.900
And then, the same thing: 1, 2, 3, 4, so it is going to be somewhere here.
00:24:32.900 --> 00:24:51.100
I know that that is my center; how do I know that?--because, if I draw a radius of the circle, this point is going to be on the circle.
00:24:51.100 --> 00:24:58.600
This point is going to be on the circle; this has a radius of 4, and this has a radius of 4.
00:24:58.600 --> 00:25:07.800
So, 1, 2, 3, 4; that is also on my circle; and that is also on my circle.
00:25:07.800 --> 00:25:17.900
Since the center is in the second quadrant...if it said that it was in the first quadrant, then you would count 4 this way and count 4 upwards.
00:25:17.900 --> 00:25:23.700
So, the circle will be tangent to these two lines, but still be in the first quadrant.
00:25:23.700 --> 00:25:36.600
I can just draw my circle like this; keep in mind: make sure that...
00:25:36.600 --> 00:25:40.600
if it is tangent, then it has to be touching at one point, and that would be the point of tangency.
00:25:40.600 --> 00:25:53.300
And then, you can just draw a radius to both tangent lines; it has to be 4 away from this tangent and 4 away from this tangent.
00:25:53.300 --> 00:25:57.700
And this would be the center, to make that 4 away from each of those.
00:25:57.700 --> 00:26:14.600
Then, our center (because again, we are just writing the equation) is 1, 2, 3...(-3,2); our radius is 4.
00:26:14.600 --> 00:26:30.500
So, our equation is going to be (x + 3)...a negative...squared, plus (y - 2)² = 4² (is 16).
00:26:30.500 --> 00:26:38.600
That is the equation for this circle here.
00:26:38.600 --> 00:26:46.300
If you ever get this type of problem, draw your tangents first; see what quadrant it is in.
00:26:46.300 --> 00:26:58.500
Figure out what your radius is, and then start finding the distance away from the tangents; and you use radius for that.
00:26:58.500 --> 00:27:01.000
That is it for this lesson; thank you for watching Educator.com.