A Level Maths Made Easy - The Equation of a Circle

hey guys welcome back to another a level preparation video this time it's going to be for the equation of a circle again the topic that came up in GCSE but there is some extra stuff as usual so you're going to be doing things like finding the equation of a circle when this Center is not at the origin if you remember rightly the GCSE form of this equation was just x squared plus y squared is equal to the radius squared whereas in this case we're going to have something slightly different so again that's what it looked like before but that's because the center was always at the origin the actual equation of a circle looks like this I know look at that disgusting where A and B are simply the center of the circle now before we get into this I've got a few things to mention so one is I got a comment uh saying basically is this every exam question that has ever come up the answer that is no so these videos are designed to give you a overview of the different styles of questions that you can be asked however we are going to go through every single past paper so I don't want to do all of the past paper questions because then when we come to do the past papers it's kind of pointless right we don't have any past paper questions we haven't seen before so that's my Logic the past papers will be done slightly later before your mocks so this isn't literally every single question next two things uh basically housekeeping the worksheet is available for download on the patreon on there you'll also find the mechanics and statistics course should you wish to purchase that as well but for the worksheets completely free click the link you can download it's nice and easy and all organized you can find all of the other worksheets as well for either download in the description there's also the a level playlist these were all of the live lessons that I did last year so they go a bit further on then we are going to get on with the a-level prep work without further Ado let's keep going so these are the differences the main differences however it says here Circle C is the equation x squared plus y squared minus 2x plus 6y is equal to 26. find the coordinates of the center and the radius of C and what you have to do here is actually find the equation of the circle and how do you do that well what you do is you use completing the square and all right that comes back so what I want to do first of all is I'm going to write the X's together and the Y's together so that equation can be written as x squared minus 2X plus y squared plus 6y equals 26 right I haven't done anything except move the minus 2x to the front which is completely fine because the order doesn't actually matter in this case and what we're going to do is we're going to do completing the square for the X's and completing the square for the Y's separately so it's almost like I'm going to split this down the middle and I'm going to join it again afterwards so before completing the square what is that well if you have an equation that looks like this then completing the square is simply plus b over 2 squared minus B over 2 and then if you have a c the C goes on the end as well in this case we won't have a c so it's actually quite small it's just going to look like this for the X's X half of 2 is 1 so it's just going to be -1 squared minus 1 squared and for y's it's going to be so put a plus there it's going to be y minus 3 so that is a 6 not a b although does look a bit like one equals 26. so we want to do now is expand these and simplify them so what we can do if you fancy being a little bit lazy which I certainly do we're going to do is we're just going to write the x minus 1 squared and then plus y minus 3 squared and then what I'm going to do is I'm actually going to work out what minus 1 squared minus 3 squared would be so realistically I probably don't need to calculate for this but just for the sake of laziness we have minus 10. minus 10. equals 26. now if you notice in the original formula there is no constant on the left hand side of the equation we need to move that over to the right hand side so add 10 to both sides and we get our final equation for the center oops not like that it's a plus between equals 36 and just so we're clear a in this case is 1 B is equal to 3 and r squared is equal to 36. so if we look at what they're asking me it says the center of the circle the center of the circle is at one comma three and for part two the radius of the circle is just the square root of 36 which of course is 6. if you're wondering why it's not plus or minus 6 it's because the radius is a length and we can't have negative lengths if I ask you to draw a line of negative three centimeters before we look at your ruler look at me and wonder what the hell I'm talking about so that's why it's always going to be positive so the first question we did was pretty much as simple as it can be however in your a-level as mocks or in your a-level exams they could chuck that at you just three marks nice and easy quick or quick the other three mark question they can ask you is they can give you the center and they can say it passes through a point four nine find an equation for C so what I like to do as usual is draw things out so let's draw our Circle first I'm going to draw over here and do my maths on the right hand side like that and this point in the middle is 2 comma five and then at this point over here which is 4 9 so again it doesn't have to be super exact there is a straight line so um actually there isn't never mind uh it just passes through that point so first things first I can actually write part of the equation because again remember the equation of a circle it looks like this I already have a and b it's just two and five respectively what I need is the radius and how do I work out the radius well have a look at this they've given it to me technically I don't need the radius I need the radius squared which actually does make our lives slightly easier now what I can do is use good old-fashioned Pythagoras's Theorem so if I draw my dotted lines going along like this the width of this triangle is the difference in the x coordinates because X tells me everything I need to know horizontally and Y tells me everything I need to know vertically so if I look at the x coordinates it goes from 2 to 4 then that means that width must be 2 and the height goes from five to nine so that height is 4. okay so in other words a squared plus b squared equals c squared that means 2 squared plus 4 squared equals r squared so that's 4 plus 16 equals r squared so that means r squared is equal to 20. so that means that is our answer we don't need to square root it because we want r squared we don't want r that's a pretty quick question and once you've practiced it you won't even need to draw it however please for the love of God they like to screw you over the lights for a curveballs at you in a level maths so drawing a diagram it didn't take me long very quick and with a pencil I've done even quicker I can then see exactly what I need to know and what I need to do okay so please please draw diagrams here we have part A which is the exact same as the previous question but then we have a couple of extra points and if you look at the marks this is a 10 mark question in total that's pretty pretty big okay pretty big indeed so let's have a look first it says we have a center of minus two and three so immediately I'm just going to write that in terms of the equation if you remember the x is minus or whatever this is so minus minus two in other words it's plus two squared plus y minus 3 squared and that equals r squared which I don't know however using what we did before we can simply work it out now because it's a multi-part question again I'm going to draw a diagram because that is a very very important thing to do you might want to draw multiple diagrams but in this case let's just draw one fairly big one because you can add to it over time that's completely fine it's not even recognizing that as a circle wow okay um and that gives me the center of minus two comma three and this is one eight so we can put it somewhere like let's say here once again all we're doing is we work out the radiuses so what's the difference in the x coordinates where it goes from minus two to one that's the difference of three right minus one zero one three to eight that's the difference of five so again a squared plus b squared equals c squared we have three squared plus five squared equals r squared so that's 9 plus 25 equals r squared so that means it's 34 yeah equals r squared so that answer is 34. four marks not too bad right not too bad Part B might be something that is somewhat more interesting to you it says show that the point three six lies on C and what you would do back in the day back in your old GCSE days is subbing the x coordinate and see if the y coordinate is spat out so if you had something like let's say Y is equal to 3x plus 2 you would sub 3 in for x and then see if that gives me Y is equal to six now you can still do that if you want in circles the problem is is you'd have to rearrange for y screw that here's what you want to do instead Sub in three and six and see if it gives you 34. brain so we have three plus two squared plus six minus three squared what does that give us well I like to do a few extra steps with this question just so the examiner can see that I'm doing something sensible and then I do equals and if you want you can use a calculator but 5 squared plus 3 squared that is 25 plus 9 which is of course equal to 34. so it is part of this equation it does lie on the circle now here's where we get it's a bit fun now this is basically a GCSE math question believe it or not it says to find the equation of the tangent to see at three six so what I would like to do is probably put three six somewhere over here and I'm actually going to erase some of this however in your actual exam you probably won't need to actually erase it and that Center is minus two comma three and it says find the equation to the of the tangent at this point so give your answer in the form X Plus BR plus C as well that is the point three comma six like so well we're going to do the same thing that you did back in gcses you're going to use the fact that the radius will meet the circumference at 90 degrees it will meet the tangent at 90 degrees so if I find the gradient of whatever this is I can do the negative reciprocal to find out what the gradient of the tangent is perpendicular lines okay if you're unsure and want to refresher on that the last video was equations of a line where we did a lot of perpendicular stuff so you can go back and have a look at that but that's kind of the concept here so I like to call it the gradient of R is equal to the change in y of a change in X so it's 6 minus three over 3 minus minus 2. so that is 3 over 5. that means that the gradient of the tangent NT you flip the fraction 5 over 3 and then you make it negative not bad now again I actually showed you this new form in the equation of a line video so normally you do y equals MX plus C Sub in M final C is blah blah blah however when you have this form where they all need to be integers it can be useful to use this equation instead y minus y1 oops equals m x minus X1 because you can get rid of fractions in one easy step again if you're unsure head back to the equation of a live video so remember the coordinate we need to use is the 3 6 not the minus 2 3 because it's the it's a coordinate that lies on the tangent not necessarily on the circle although in this case it's both so y1 is six so y minus 6 equals the gradient minus five thirds x minus three and immediately because I see here it says they will have to be integers I'm going to times through everything by three so we get 3y minus 18 equals minus five x minus 3 expand the brackets three y minus 18. equals minus 5X plus 15 and now lastly but certainly not least move everything over to the left hand side and we're done X goes first because it just says that in the question so plus 5X plus 3y and then we have minus 18 minus 15 again if you would like you can just slap it on a calculator oh cancel that minus 33. minus 33 equals zero five marks in my opinion not that bad once you get used to it the good thing is is that whenever they ask you to find the equation of a tangent they have to give you a certain amount of information ninety percent of the time it's just the coordinate where it meets the circle if that's the case easy five marks Because the actual layout is the same gradient of radius negative reciprocal sub into the formula done those two steps all the time here's yet another style of question part A in my opinion is very very generous because if you have a look they give us everything we need so for part A all we need to do is shove in the A and B so minus 2 squared plus y minus 5 squared and don't forget it equals r squared not R so 7 squared is 49. just that's the only bit that I think you might miss up on in which case you get one out of two so still pretty generous Part B you might notice there's six marks and you might be thinking Jesus Christ but if you have a look at what the question States it says the line Y equals 3x minus 1 intersects C at two separate points find the exact coordinate so if you remember the discriminant video we did before that means it's going to have two solutions so what this effectively means is when a line intersects with another curve or a circle in this case then that means the X and Y values are the exact same so what we can do is use simultaneous equations to work this out so all I'm going to do is take this entire equation and because it already says y equals I'm not going to rearrange it I'm just going to shove it in for y here and it's going to give me a quadratic simultaneous equation okay so two values for X which I can then use to work out two values of y now that might sound really tedious and kind of overwhelming but I have shown you before how to solve simultaneous equations on that white and black calculator that I thoroughly recommend there's a link to it in my in the description by the way in case you just want to pop it open and buy it now it's so useful and I would say it's damn near essential for statistics but anyways let's continue we have x minus 2 squared and then y we're going to sub in everything so 3x minus 1 minus 5 squared equals 49. now you can probably see already it's a quadratic simultaneous equation so what do we need to do we need to expand everything and then simplify but just before that just to make the algebra a little easier I'm going to just uh simplify this bracket so minus 1 minus 5 is a minus 6 squared plus 49 perfect now let's actually do the expanding expanding quadratic double brackets I'm hoping you're completely fine with uh I do go through it a bit quickly so for example x times x would give me x squared and then we'd have minus 2x minus 2x that's minus 4X and then plus four if you're struggling with this let me know in the comments and I'll just type it out for you okay and then obviously then you should definitely go practice this and for this one it's going to be 9x squared then it'd be minus 18x minus another 18x that's minus 36x then plus 36 equals 49. let's simplify what's on the left hand side so x squared plus 9x squared is of course 10 x squared then we have minus 4X minus 36x that's minus 40x then we have plus 4 plus 36 is 40 equals 49. now of course when we solve quadratics remember you need to move everything to one side it needs to equal zero so that's minus 49 from both sides real quick so we have 10x squared minus 40x and that's minus 9 equals zero now let's use our calculator to work out these values so you go to menu go down twice and write once it says equation funk equals we're solving a polynomial the simultaneous equation works for linear equations only so it's kind of useless for us so two polynomial degree it's going to be two because it's squared right if we look at the quadratic polynomial degree 3 is a cubic so something that has X cubed in it and then four is quartic which you won't use unless you're doing uh further maths so two then we sub in the numbers we have 10 equals minus 40 whoops equals and minus 9 equals we get two pretty disgusting numbers but it says find the exact coordinates that's a bit annoying because what that means is we need to give it an exact form so essentially we have to leave it like this and you'll find that in a lot of kind of a level questions you get pretty disgusting equations instead of or you should technically write and because it's two separate equate um coordinates yeah probably didn't need to press equals to work that out but hey ho okay and now I like to split the thing down the middle so we need to work out now the Y values and the really useful thing here is we have our equation already so all we're going to do is sub in those two x values in to get our respective y values so let's start with the one on the right because what I can do is use the good old-fashioned answer button so this is my answer now go to menu go back up to calculate or just press the number one on your calculator we can do three times answer minus one and if you're really paranoid like I am you can even put this all in Brackets so you do it in the right order that gives negative 100. that's pretty useful and for the other one we obviously can't do it we're just gonna have to do three and then we're gonna have to type this whole thing in 20 plus 7 root 10. minus one we get that you know what that that value is actually making me very suspicious I'm going to type in this first one again minus 100 I just realized it shouldn't really be a whole number so all that's different is this thank God I listened to my gut okay so it's 15 minus so get rid of this 50 minus 21. root 10. over 10. so our two coordinates for a and b it doesn't really matter which way around we've we put them because they haven't actually given us any kind of indication as to what A and B are as in is a a lower x value is b a higher x value so in this particular case we can't say which ones a and which one's B with 100 certainty in some cases they will make it obvious they might say a is something and B is something else like a is the x-intercept B is the Y intercept or something like that so let's write these out just so the examiner can have no excuse to take away our hard earned marks and yeah in a level we do get very ugly answers sometimes but that is the way it is 10 over 10 50 minus and that's how I leave my answer at the bottom one final style of question here it says the circle has a center five and the K where K is a constant the line is a tangent to the circle touching C at this point find an equation for C so in this case you know the kind of a previous style of question was we had the equation of the circle we had to work out the equation of the tangent now we have the equation of the tangent and we need to work out the equation with the circle so effectively we're going backwards which is completely fine so first of all again let's draw our little diagram that wasn't actually a bad Circle myself so 5 comma K I know that this point should technically be on the left hand side but I also don't care I prefer to draw it from left to right I don't know why and we have our tangent like this which has an equation of two X plus one now let's use all of the information at our disposal so what would we normally do or normally we would find the gradient here and use that to work out the gradient of the tangent and then we're basically done well we actually follow the same steps but kind of in reverse we don't know the gradient of the radius but can we work it out the answer to that is absolutely we can because we do the same thing but backwards this line here the radius line let's call it is perpendicular to the tangent but that also means the tangent's perpendicular to the radius line so the gradient of the tangent is 2 right it's the number in front of X which means the gradient of the radius flip it over so 1 over 2 and make it negative now we need to work out K because we need to know the center of the circle so what can we do to work that out well how would we normally work out the radius the gradient of the radius we would do the difference in y divided by the difference in X let's do that right now so difference in the Y coordinates is 7 minus k over 3 minus 5 and that equals minus a half can you see the point so now I'm going to actually simplify this slightly just to make my life ever so slightly easier I'm going to times both sides by minus 2 to get rid of the fraction so 7 minus K now you can believe me if you want but that's just going to give me plus one however let's use our calculator to verify we have minus a half times -2 1. so now if we kind of swap this round we have 7 is equal to 1 plus K so K is just equal to the number six we're almost done now we actually know what the equation of the center is the equation of the center or the coordinates here are going to be 5 comma six so in other words we have x minus five squared plus y minus 6 squared and what does it equal equals the radius squared how do we work out the radius we do Pythagoras's Theorem so it's limiting the x coordinates it goes from five to three so technical is minus two but if you remember with Pythagoras you square it so you don't actually care also we're using this as the length of a triangle which means it shouldn't have negative components and then six goes to 7 which is one so that means r actually let's let's do it the other way around 2 squared plus 1 squared equals r squared so four plus one equals r squared which means r squared is just equal to 5. that is our equation of the circle for five whole marks not that bad I hope so that's everything we're doing today again this isn't literally every single question ever but this is pretty much every style of question that's ever come up for the rest of the questions we're going to save those for when we go through every single exam paper in existence which is going to be quite fun we're going to do that towards your mocks so we're gonna go through all the year 12 papers okay hopefully that makes a bit of sense but yeah I'll catch you in the next one but remember the link to the a level playlist in the description the link to the worksheets in the description and of course check out the patreon for the mechanics and statistics content as well