00:01
So in this first problem, what we're being asked to do is to find the equation of a circle whose endpoints are of the diameter happen at negative 82 and negative 4 .4.
00:09
Well, first off, let's talk about our standard form for an equation of a circle.
00:13
It looks like this.
00:14
It's x minus h quantity squared plus y minus k quantity squared is equal to r squared.
00:20
Well, the center of our circle is going to be the ordered pair, hk, and r represents the radius.
00:27
So in order to write our equation, we're going to need the center as well as the radius.
00:32
Well, let's go ahead and start by plotting these two points.
00:35
So negative 8, 2, that's left 8 and up 2 units.
00:39
And negative 4 -4 is left 4 and up 4 units.
00:42
So that would be the diameter.
00:44
Well, if that represents the diameter, the midpoint is going to be halfway between.
00:49
Well, how do we get from negative 8 from the first point to the second? well, we go up 2 units and we go over 1, 2, 3, 4 units.
00:57
So if we were to cut that in half, we would only want to go up one and over two.
01:02
So if we go up one and over two, notice that the center of the circle would be at the ordered pair, negative six, three.
01:09
So now we know our center.
01:11
It's negative six three.
01:12
So now what we have to do is find the radius of the circle.
01:15
Well, notice that if we had these two points, negative six three, and let's go with this first one, negative eight, two.
01:22
We just need to find the distance between them.
01:24
So what we're going to do is use our distance formula, and that will tell us the rate.
01:28
So the radius is going to be equal to.
01:30
Remember, for our distance formula, first we're going to find the change in our x values.
01:35
So we're going to have negative 8 minus negative 6 quantity squared.
01:39
And we're going to add this to the change in our y values, so 2 minus 3 quantity squared.
01:43
So now we just need to simplify.
01:46
Well, these two negatives become positive.
01:48
So really it's negative 8 plus 6, which is equal to negative 2 and negative 2 squared is 4.
01:54
Then we have 2 minus 3, which is negative 1, and negative 1 squared.
01:57
Squared is 1.
01:59
So we have 4 plus 1, which is 5.
02:01
So now we know the radius is equal to the square of the 5.
02:04
So now that we have our center and that we have our radius, we can go ahead and find our equation.
02:09
So for the equation of our circle, we're going to have x minus our h value, negative 6, quantity squared, plus y minus k, which is 3 squared, all equal to our radius, the square of the 5, square.
02:22
So now we can just simplify a little bit.
02:24
Well, these two negatives make a positive, so we're going to have x plus 6 being squared, plus the quantity of y minus 3 being squared, and then we have the square of the 5 squared, which is simply just positive 5.
02:36
So perfect.
02:37
Now we have the equation for our circle.
02:40
So in part b, they want us to state the center, which we did, and the radius, which we did, and now we can go ahead and sketch this graph.
02:47
So in this particular case, notice that we have our two endpoints here, and here's the center for our circle.
02:52
So now what we can go ahead and do is we can find a couple of other endpoints on the diameter.
02:59
Because remember, if we go up one and over two, we can go down one over two to find another endpoint, or up one and over two to find another.
03:07
So now we can go ahead and draw our circle.
03:10
So now we have our circle.
03:12
Okay, so now let's take a look at number two.
03:15
Or what you're calling number four.
03:17
You want to write the equation in the circle in standard form, as we mentioned below or above, by completing the square.
03:23
And then we want to find the center and radius.
03:25
So in this case, you have the equation 8x squared plus 8y squared, getting subtracted by 24x...