Three Mistakes Students Make With Circles

all right let's go ahead and talk about three mistakes that students commonly make when dealing with identifying the center and the vertex of a circle so in this equation or in this example we have all three equations of a circle now there are two forms of equation of a circle that we need to make sure we understand so we can identify why these mistakes make sense or why they happen Okay so the two equations that I have is X S Plus y^2 = R now this is going to be when we have a circle with a center at the origin and here we have x - H2 + y - K con 2 um equal r s and this is going to be when we have a circle with a center outside of the origin now the X and Y represent all the points that are going to make up your circle R is going to represent the radius that's going to be the distance from the center to any point that lies on the circle and in this case H and K are going to represent the center of the circle so obviously if that was 0 0 like if the center was 0 0 we wouldn't need to subtract by um Ag and gay and that's why you get this formula all right so let's go through the mistakes the most common ones students make all the time the first one students will say Mr mclogan the radius is going to equal to a 25 and no like we love when we see a square number at the end but ladies and gentlemen the radius is not going to be 25 remember guys the formula is X2 + y^2 = r^2 so it's x^2 + y^2 equals what number squar equals a 25 that's going to be a 5 squ so therefore we can say the radius is going to equal a five not a 25 so let's not make that mistake the next one happens again for everyone so don't feel stupid it happens all the time I even make this mistake a lot of times so take a deep breath go slowly and don't make this mistake so a lot of students will understand ah okay radius is three not going to make that mistake again right you usually do it once and then you're like I'm not doing that again but then this next one this one's get confusing so the re I like to actually rewrite this equation so you can see it is with additional parentheses now the reason why I wrote additional parentheses is because I really want you to identify what are the values of H and what are the values of K because remember this Center is going to be H comma K so whatever those values are that is going to be the center so a lot of students will assume that the center is going to be at a 5 comma -3 because Mr mclogan I see a five I see A3 that's your H and K we're done but remember guys it's about the formula it's x - H + y - k^ 2 x - H Yus K so it might be helpful to even put parentheses in this formula so you can see what exactly are going to be the values of H and K right this is incorrect the values of H is going to be A5 the value of K is going to be a posi 3 so the center in this case is actually A-5 and a posi3 and again again if I simplify this a lot of students are like well how do you go from a plus to a negative well remember x minus a negative is the same thing as adding right so therefore you can see how that happened now on the last example a lot of students will make a mistake here and the reason why they make a mistake is they say well what do I do with these twos in here now notice go back to the formula I provided you the formula over here did not have any coefficients of your X and your y s or the quantity x - H2 and Y - k^ 2 like they didn't have those so what do we do these twos well we got to get rid of them so how do we get rid of the twos well divide by two on both sides now a lot of times students don't want to do this because I see 16 and they say oh 16 R squ I know the radius is going to be a four that's easy crap but now what if I divide by two I'm going to get eight and that's not going to give me a uh an integer radius but guess what that's okay all right so now again we recognize here now this one doesn't have anything to subtract by right it's xus nothing right so therefore we can say it's going to be a zero so the center in this case is going to be a zero and then here remember it's the opposite value so therefore it's going to be A7 and now the radius we just need to say well what is what number squared equals 8 so we could say R 2 is equal to an 8 take the square root of both sides now we don't want to use a decimal equival equivalent like 2828 4271 what we want to do is simplify this radical you could leave it a square of eight but typically we like things simplified so I can bre rewrite that as a four time 2 take theun of 4 which is going to be a 2 of two so therefore my radius in this case is going to be 2qu of two those are three mistakes you don't make them