okay well now we're into the last part of this lesson and it's the circle you say how does the circle relate to all that other stuff well you're going to see it's the distance formula reworked here now i'm going to begin with you at the center at the origin here so let's go ahead and we've got our coordinate system set up let's put a dot here to mark the center now i want you to visualize with me some circle around here let's let's go grab a point on that circle just one point you can you picture the whole circle through here all right let's draw from the center out to the edge that'd be a radius measure okay so let me just mark that here this would be a radius now earlier we called that a d for distance formula but now we're just calling it an r for radius and what i'm looking for is how far is it how do you get that r value you say well i can do that pythagorean theorem see if i leave this point and just call him 0 0 because he's at the origin that's what we're looking at and this is just some generic x and y and actually this is let me make that prettier this is one particular one so i'll just call it x1 and y1 you say how would you find that distance across there because you'll remember that the distance formula i'm just going to copy it over from what we had previous in the notes it says x2 minus x1 squared and y2 minus y1 squared and take the square root of that that sum okay so now i'm changing out my letters look we've got r and then instead of x1 or excuse me instead of x2 and x1 i've got x1 and 0. so just to make it look pretty i could it doesn't matter which one you write first but it'd kind of make more sense if i'd write the y's and the x's first so now we're looking at x1 squared and y1 squared now we're getting close to the answer here let's undo this square root because we don't want to know the distance across here let's just go with r squared you say what we'll square both sides of that thing and that'll get rid of that radical out of there so that would be an x1 squared and a y1 squared but you know what i don't need it just for one little x1 and y1 watch carefully here i need it for any of these around here so here's here's an another point here's another point and another point and you can see all these little radius measures out here so in other words it's for any and all of these x's and y's around here so i can get rid of this little subscript and just say we're opening up to all of them and there's a formula now most of the time most students will recite it as x squared plus y squared equals r squared and that's for that's for the formula for a circle centered at the origin now while we're there so hold on to that thing while we're there let's look at it let's let's look at one more screen here where we center this thing at another point called hk see we were centered at the origin back here hold on i'm going back we were centered at the origin which is zero zero now notice where these look look at where these numbers are coming in let me see if i can highlight this for you see right here there's my zero and there's my zero you see in those well i'm going to change that and make that an h and a k you say why h and k well in math those letters usually represent kind of some kind of center they're not used for much else so let's go back over here and say let's center that thing at h k let's move that center point now hold on i'm going back we're moving that center point back off somewhere over here call it h k now put your dot out there draw your radius draw that circle around there now remember that we had x squared plus y squared equals r squared but now since i've got this h and k see this was a while ago this was x minus zero without taking you back another slide but see now i don't want that that zero anymore i want to replace that with an h k h and k so now my formula looks like x minus h squared and y minus k squared equals r squared and that's the formula so now what i want to do is just practice that formula with you so let's go look at several these are going to go fast exercises here so let me write the formula right up above here just so we've got it says x minus h squared and y minus k squared equals r squared so i'm trying to determine what's the center and the what in other words what is h and k on each one of these well let's see what we've got now follow with me here x minus h well the h has to be a 1. now be real careful here this is reading y minus a negative 4. so come come back over there that's y minus k k is negative 4. now that's going to take a little bit of getting used to but a lot of times when i'm working with students they just say it's the opposite of what you think so this looks like a negative one that means this is a one over here this looks like a four this is the opposite and you can do that over and over and over and over and over okay so let's let's try it on this second one here the center is going to be at and you would say negative 2 7. let's try this one the center is going to be at negative 5 0 because there's an understood y minus 0 squared just like i'm just going to repeat it x minus a negative 5 squared so when we say negative 5 0 and then this last one the center which is at h k so get practice writing centers at h k this one's going to be at negative 3 2. all right hopefully you're getting that really well now the next thing is what's the r value so the radius which is r is going to be and you have to look at the formula here it's going to be the square root of this number so square root of 25 5. the radius for the second example square root of 9 3 radius for this next example square root of 16 4 radius for this last one square root of 7 yeah just square root of 7. now if i were trying to draw it and trying to put it out there you wanted me to go to this point negative 3 2 and then count over then i would have to grab my calculator and this would be approximately we don't we don't need it for this exercise here but it'd be approximately about 2.6 wait a minute there we go 2.6 you say how would you draw one of these well why don't we pick on this bottom one here it won't hurt anything so let's look at it so first thing i want to do is clean off some space it would really help if you had a decent piece of graph paper but i'm just going to make it work just to make sure you've got the concept of how do you graph these well so we're looking at this fourth example here so we've got a center at negative three one two three one two so there's there's my center point now what i want to do is count out square root of 7 which is approximately 2.6 so bear with me here i'm going to go one two and just a little bit more ah somewhere right in there just make myself a little dot again and i just do this north south east and west so now i come over one two and just a little bit more so it's going to be ah this pencil wants to do something different but i'm just going to make myself a little dot there and then i come down one two and just a little bit more not quite three about two and a half close to it and then come over the other direction one 2 and just a little bit more 2.6 all right and then do the best job you can to draw a decent oh yeah i know i don't know that this one is perfect but anyway that's good enough and that's how you draw it all right now we want to take this one step further see i want you to think with me what if we were to take one of these and foil it all out you're like what foil all that stuff out why would you want to do that well sometimes it comes all foiled out and you've got to put it back into this form right here and that's where we want to go next so watch with me as we go to the next example i've got 2 just for practice here x squared y squared 12x minus 6y minus 4 is 0. now the first thing you want to do is just copy that thing down in your notes and when i copy it down let's clump the x's together all right so that takes care of that term in that term let's skip a little bit of space just enough to put another term in and let's put the y's together there's the y squared and i need a minus 6y skip a little space for another term put your equals and then this term right here we're going to move it to the other side so that's going to make it a positive 4 over here got that okay so my next step is to complete the square let me write a little note here complete square all right so how do we do that well we take half of the 12 and square it so let me draw a brain over here this is your brain and you're saying let's take half of the 12 by the way that's 6 take the 6 and square it so 36 is my magic number so i'm going to put a 36 right here now what did i just add to the left hand side of this equation you say 36. well i got to keep it balanced so i got to put 36 over here once you do to one side you gotta do the other side perfect so let's pick another number here let's pick on this one so what is half you don't have to worry about the negative the negative will take care of itself what's half of six three squared nine if you need a brain on that one half of the 6 is 3 then you square it you get the 9. so you're going to put 9 on the left you're going to put 9 on the right very nice now we want to look at these three terms the clump of x plus that constant we added and i want to factor it now factoring depends on how how well you can factor but that should be x plus 6 squared you say how do you get that so fast well one thing you practice factoring the other is you write down the letter copy down the sign and what was half of that 12 right there let's do it again this is going to be y minus 3. copy down the letter copy down the sign whatever half of that was that's part of the completing the square process on the right you just get whatever you get that's 40 49. so now we have it cleaned up so what's the center you'd say negative 6 3 and the radius 7 is the square root of that good and if you wanted to we could graph it but yeah so that's what we wanted to figure out on that one let's do this this last one here this on the screen for practice okay so a little bit shorter equation here so the rule was you copy down the x's and that's all the x's we had leave a little space not really you have to but then y's we got y squared and y practicing just leave a little space and move the half over so i want you to understand there's there's nothing to do with the x if you want to you could write it as x minus 0 squared but you don't have to do that let's focus on this completing the square over here now understand that that's a 1 right there so my brain is saying i'm going to take half of the one which is just a half and we're going to square it now don't make this hard here when you square it you square the top 1 squared is 1. 2 squared is 4. so there it is so you're going to add a fourth here so you need to add a fourth to the other side so now in terms of factoring you've got copy the letter down copy the sign down and what was half of that one half don't forget that little square part of the formula what is a half plus a fourth in other words two fourths two four same as a half two fourths and one fourth makes three fours and there's my formula so you say well yeah let's do the center do you have to write the the minus zero no again if you want to leave it that away or if you'd rather just call it x squared it doesn't matter the center is at h k so that was a zero remember it's the opposite of what you think here oh yeah okay now the radius is take the square root of this thing oh my it's a fraction don't worry about that square root of the top and square root of the bottom and that's how you go back and find the equation this is this is the key piece once you got that the other you can figure out all right i have one last example in this lesson we're looking for the equation of a circle with end points of the diameter diameter we haven't been talking about diameter yet diameter at negative 1 3 and 3 11. so let's let's kind of visualize what's happening here let me grab my graph paper so we don't we we don't have to be perfect with this we're just trying to get a concept in our head so negative 1 3 is going to be oh i don't know i'm just kind of saying somewhere right in there let me just mark it negative 1 3. and 3 11 let's see so 1 2 3 way up here i don't know it's somewhere right in there nope so we'll mark that call that 3 11. all right so i need a circle that goes through there now the center now think about this with me this is a key point the center of my circle well hang on a second if this is the diameter and that's the midpoint i just set it that's the midpoint so the center is at the midpoint well let's go find the midpoint so how do you do that you add them up so take the negative 1 and the 3. so take the x values add them up divide by 2. take the y values 3 and 11 add them up divide by 2. what does that give me 3 minus 1 is 2 2 divided by 2 1 14 divided by 2 7. there's my center now remember remember the letters we were using that was h and k just so we can practice okay well anyway where were we back over here we were in the process of drawing this thing this was the diameter so now we need to come over and picture with me there's my circle and i've got my h in my k all i need is to know how far that radius is well the radius let me make a note here the radius is half of the diameter oh but i don't know how long it is doing but i could go find it so the diameter this whole length across here would be the distance formula let's call it well this is perfect d for diameter the distance across there is now remember that's x2 minus x1 so subtract my x's there's 3 minus a negative 1 squared and then y2 so 11 i'm just calling this y2 11 minus 3 squared all right let's see what that is three plus one that's four we're going to square that 11 minus three is eight and we're going to square that all right let's see that'd be a 16 and a 64. and let's see that makes 70 80 so square root of 80. that's the distance across that whole thing now can we simplify that we really do need to break that thing down a little bit that's going to equal what is i see a 4 goes in there 4 times 20 or you could say 16 times 5 so that would be 4 square roots of 5. so this distance across here where is it is 4 square roots of 5. so the radius would be half of that well that would be two square roots of five but my formula because remember here comes my final answer where am i going to put that how about up here my formula says x minus h squared plus y minus k squared equals r squared so my formula is going to require an r squared so square that thing 2 squared is 4 square root of 5 squared is 5 5 times 4 makes 20. okay now come over and write your answer x minus the h that'd be 1 squared and y minus the k in there and put your r squared and there is the puzzle wow that connected distance formula midpoint circles all together fantastic work very good now some people ask me the radius is half the diameter and so i went and found the diameter other people have said look the radius is just the distance between this point this point right here we found as 1 7 and 3 11. hey that'll work too just go find that distance instead of taking this one and cutting it in half it doesn't matter it'll get you the same answer all right congratulations that was a good lesson for you