additional maths today we're going to be looking at the equation of a circle okay we're going to be looking at questions like the one there which says find the center and radius of the circle with the equation x squared plus y squared plus 8x minus 10y minus 9 equals 0. okay that equation defines a circle on a grid but before we get into that question we need to look at some prerequisite knowledge some knowledge you need to have already in order to be able to solve that problem so at gcc level you're expected to know how the equation of a circle with a center at the origin okay so if i have a circle and here is a terribly drawn circle but imagine that was a circle with a center at the origin and with a radius of r so the distance from the origin to any point on the circle is r okay then there is a there is an equation which defines all these points on the circumference of this circle all those coordinates okay just as there's an equation that defines all the points on this straight line and it will be y equals mx plus c for some m and some value of c so there is an equation which defines all these red points on the circumference of this circle okay the key is to start with a specific point this point here okay let us call it the point x y so it has an x coordinate of x and a y coordinate of y naturally okay how could we define it with the information we have here well it's quite simple we use pythagoras theorem we know that pythagoras theorem will hold for this triangle okay no matter where i put my my coordinate along this circumference i can create a right angle triangle which has an x coordinate of x and a y coordinate of y and therefore this this side of this triangle is x it's the distance i've gone from the origin to the coordinate of x the x coordinate of x i've gone right x and i've gone up by y so think of an equation which holds true for that triangle and that is that x squared plus y squared is equal to r squared and that is the equation of a circle with the center at the origin and a radius of r so that will always be true so if i had this graph and this coordinate there was 7 and that was seven and that was minus seven and that was minus seven so the radius is seven that would have an equation of x squared plus y squared equals 49 or seven squared if you prefer writing it like that okay so that's a center at the origin now let's look at ones with a center away from the origin so let's look at a circle and that i promise you is a meant to be a circle with the center which has coordinates a in the x direction and b in the y direction and which has a radius of r okay we do exactly the same process we try and create an equation using pythagoras so the radius is r and let's take a general coordinate on the circle which has an x coordinate of x and a y coordinate of y all we need to do is find this distance and this distance and create a right angled triangle and then create an equation using pythagoras for that well the horizontal distance is the horizontal distance from an x-coordinate of a to an x-coordinate of x that distance there is the difference between x and a so it's x subtract a will give us that distance and similarly in the y direction this distance is y subtract b it's the difference between y and b okay and therefore the pythagorean formula which holds true would be x minus a all squared plus y minus b all squared equals r squared that is the formula for the equation of a circle which has a center at a b and a radius of r okay so copy that down make sure you have that information handy because that is how we're going to define the equation of a circle when we have it in that form we then know where the center is and what the radius is so i'll just do a quick example based on that if i have this graph and let's draw a circle which just it's meant to just touch the y-axis okay so just barely touch the y-axis and it has a coordinate at the center of minus 3 8. then the equation would be x minus the x coordinate so x minus minus 3 which becomes x plus 3 all squared plus y minus the y coordinate which is 8 all squared equals the radius squared now because it just touches the y-axis can you think about what the radius would be the radius is this distance well done if you managed to work out that was 3 and therefore this would equal 3 squared or equal 9. so that is the equation of the circle that i've drawn okay so that's prerequisite knowledge you need to have now what we're going to do is we're going to try and find the center and radius of the circle with this equation here so what i really want is i want to be able to rewrite this equation in the form x minus a l squared plus y minus b all squared equals r squared and the way to do that is by using our our amazing friend completing the square okay firstly i'm going to rewrite this with all the x parts together and all the y parts together so firstly i'm going to rewrite this as x squared plus 8x and then y squared minus 10y and then minus 9 equals 0. and i'm going to complete the square for the x squared nate x i'm going to complete the square for the y squared minus 10y and see where i am at that point so the x squared at 8x becomes x plus 4 all squared that will give me x squared it would give me 4x and 4x which will give me 8x and it'll give me 16. i don't want 16. so i subtract 16. y squared minus 10y becomes y minus 5 all squared that also gives me 25 i don't want so i have to take away 25 and i also have minus 9 equals zero now what i want is i just want the two brackets squared on the left side and then any numbers on the right side to help me find the radius so i will keep my x plus four all squared i will keep my y minus 5 all squared and i'm going to add 16 add 25 and add 9 to both sides and that will give me equals 50. so have a thing so if that's true what is the center and what is the radius of this circle since we know the formula if the center is a and b it'll be x minus a all squared plus y minus b all squared then here the center will be coordinate minus 4 5 okay a way to think of it is just whatever makes that bracket zero whatever value of x makes that bracket zero whatever value of y makes that bracket zero that will be your x coordinate y coordinate of the center and the radius since it'll be x plus all squared sorry x minus a l squared plus y minus b l squared equals r squared so here i know that r squared is worth 50. so the radius will be root 50 and it's much nicer to write that in simplified third form so that's root 25 root two which is equal to five root two okay five lots of root two so the equation that i had at the start is defined as a circle with a center at -4 5 and with a radius of 5 root 2. okay now i want you to have a go at the next one so find the center and radius of the circle with this equation that i'm going to give you okay so this one is going to be x squared plus y squared minus 4x plus 6y minus 3 equals 0. pause at this point do some lovely algebra complete the square create it into the form you want and then write down the center and the radius of this circle okay i'm going to go through the answer now if you've done this right you would have completed the square for the x squared and the minus 4x to be x minus 2 all squared minus 4 and you have completed the square for the y squared and 6y to be y plus 3 all squared minus 9 and you've also got the minus 3 equals 0. then putting it in the right form you will have x minus 2 all squared plus y plus 3 all squared equals 16. okay so you add 4 add 3 and add 9 to both sides and that's what you get and therefore the center of this circle is at the point 2 minus 3 and the radius is simply four okay because four squared is 16. if you manage to get that right superb well done okay what you should do now is you you should go and try and do the questions from exercise 5.2 okay so the second exercise in chapter five and try and do as many of the different questions you can going on to the the complex ones towards the end of the of the exercise which are more sort of exam type ones develop your fluency make sure that this this skill being able to understand the equation of a circle is natural and free-flowing for you okay so that exercise is within this book superb book okay and i'll see you in the next video enjoy