welcome to the first video of grade 12 analytical geometry we are going to start this lesson by looking at the equation of a circle so you know in previous grades I can grade 11 you had a parabola for example which typically had or one of its formulas looked like that or you could have had a hyperbola which typically had a formula like that well a circle is another type of graph and it has its own equation which is the following what I'm now going to do is explain what the a b and R represent so a represents the x value of the center of the circle and I'll show you an example just now of what I mean b is the Y value or the y coordinate of the center and then R is the radius of the circle now we're going to put that into practice so if we had an uh if we had a circle that was given as follows which has the following center point so you see the center coordinate is 2 and 4 then the equation of that Circle oh and I must also give you the radius let's say the radius is 7. so then the equation of the circle would be the following you would make your two brackets with a plus in between them and you would have X and Y then your Center is two so we'll say minus two and then the other the Y Value Center is negative well it's four but we always use the opposite and then and I'll explain I'll show you a few different types just now and then the radius is seven but this is where I've seen a lot of mistakes this part of the formula is supposed to say radius squared so it's 7 squared which is 49 so that is the equation of that Circle let's look at another one so now the circle is in this quadrant so we can still use the circle formula of course so it'll be so now your x value of the center is minus two so what we're going to say instead is plus 2. so this formula always goes in opposites then the center of the circle for the Y value is positive four so in the circle formula it'll be negative 4 and then the radius is still seven so we'll still say 14 it's in this quadrant over here then the equation will be X okay I'm just going to quickly fill in the template so it's going to be X plus 2 y plus 2 and then the radius is still 7 so that'll be 49 so I'm sure you get the idea of what we mean by the center of the circle just remember to use the opposite coordinates so here we have to determine the equation of the circle with the following information at the center is located at the origin and the circle has a radius of 5 so we could use the circle formula again where this is X this is y then the center is at zero and zero that's what we mean by the origin so you could say minus zero here and minus zero here whoa what happened there like that over there and then the radius is 5 so remember that this part mustn't say five it must say 5 squared which is 25. now you don't have to put the minus zero because x minus zero is just X and so you eventually going to end up with an equation like that so this is typically how teachers would have introduced circles to you so just remember that that is a circle but it is located at the center so on a diagram it would be something like that it hasn't moved left right up or down so here's quite an interesting one we have to determine the equation of a circle where they have given us the coordinates of the diameter so guys what you need to realize is that you have been given a circle or you have been given the diameter okay now the diameter always goes through the middle and they've given you the two coordinates of or they've given you the coordinates of the diameters they've given you a for example and they've given you B so if we know that a diameter goes through the center then surely we if we could find the midpoint then that would be the center of our Circle so we can take the coordinates of A and B use the midpoint formula and find the center so the midpoint formula is the following and so we could take the coordinates of A and B and do the midpoint formula so it'll be 7 plus 11 over 2 and then 10 minus 2 over 2 and that's going to give us 9 and 4. so that is the center of the circle so in the circle formula we can already fill in the center point which is the x value is 9 so we put a minus and then the Y value is four so we can put a minus four all that we need now is the radius now remember the radius is this length over here so if we could work out the length of a b using the distance formula then we could just divide that by 2 because then that would be the radius so we could use the distance formula first and then we can just do the distance formula between A and B so I'm going to start with a chord and that so it would be the square root of 7 minus 11 plus then it will be 10 minus minus 2 so I'm just going to say 10 plus 2. you type that all in on the calculator and that will give you a value of I'm going to leave it in third form just to make sure we keep as many decimals as possible so that's going to be 4 root 10. so the length from A to B is 4 root 10 so the radius will be 4 root 10 divided by 2 and that's just going to give us 2 root 10. then we know that this part of the circle formula is the radius squared so I'm going to square this value now so that'll be 2 rooting and then I Square it and that's going to be 40 so I can put a 40 over here notice how I left it in third form because if I went and got decimals over there then I would have had horrible decimals to work with and it would just make that it wouldn't round up nicely to a 40 at the end here's one that says that the center point is -2 and 7 and the radius is eight that's quite a nice one so we know that the center is -2 so in the equation it'll say plus two then this will say minus seven and then the radius is 8 which means that we will say 64 over there because 8 squared is 64.