A Level Mastery - ULTIMATE Circles Question

what's up guys welcome back to another kind of a level Mastery video so today we have a very interesting coordinate geometry question that is 10 whole marks in one part and it really does summarize pretty much all of coordinate geometry with the sole exception of when you have to work out areas of triangles and things like that so it's pretty much the entire chapter Allin one so we have a straight line with equation y = 2x - 3 is a tangent to a circle with the center at C uh to3 determin an thir from the radius of the circle so once again with pretty much all a level maths questions barring just pure algebra you want to draw a diagram so I'm going to draw a big circle let's quickly discuss all the different ways that I'm going to help you in your GCC or a level maths careers first of all I want to talk to you about the YouTube members this is a really cost-effective way of getting tutoring so I'm going to do weekly live streams for GCSE and AEV maths and that all comes for £18 a month and you get at minimum an hour of tutoring per week in these live streams at the moment they're quite small so you basically get tuition for about £450 an hour which is really really cheap the second thing is the patreon for those of you doing a level maths I have already completed as stats on my patreon it is a full Clause with worksheets mark schemes videos I answer all the questions we're going to do exam papers and everything that needs doing we're halfway through mechanics as of yet and lastly there's tuition I still have places in my group classes for a level maths and GCSE Maths for GCSE it's £30 an hour and for for a level is40 per hour so if you want to let me know if you're interested just email me or join the Discord where I'm very very active all the time anyways back to the video and then I'm also going to draw a tangent and then just move this over okay again it doesn't need to be super accurate or anything it just needs to be a representation of what we have because human beings brains are very very visual we're very good at recognizing patterns so we have the center here is 2 -3 and we have this tangent here y = 2x - 3 now you might look at that and think well what information do I have well what I would first focus on and this is really how you get that a star grade is what do you need okay that's what we're thinking so in a perfect world how could I determine the radius of the circle given uh not this information but as much information as I could well what is the radius is the distance from the center of a circle to a point on its circumference so in other words I need a coordinate on the circumference of this circle if I have that then I can use the formula which is basically just Pythagoras to work out the distance which would be the radius that's the intended solution so now I'm thinking okay I need another coordinate well I can't really get a coordinate on any point on this circle except the point where it crosses the tangent right because I have I haven't got enough information I can't work out this point over here because how could I work that out I don't know so I'm going to put a point there and I don't know the coordinates of this point so I'm just going to call it X and Y because I don't know the coordinates of it so now I have in my mind a picture of exactly what I want to do I want to work out the coordinates somehow and then use that Pythagoras equation to work out the length of the line so okay what information do I have well if you think back to questions when you have to work out the tangent what do you normally do well you find the gradient of the radius and then you do the negative reciprocal of it because because of circle theorems they always meet at 90 degrees so if I draw a little dotted line along here that is a perpendicular line so I can work backwards to work at the equation of the radius as in the equation of the line and then using simultaneous equations work out what XY is so let's have a look shall we well the gradient of this line is of course two which means the gradient of the inverse so the gradient of the the inverse as I was saying is Well the gradient of the radius is the negative reciprocal which is negative a half cool now what do I need to actually work at the equation of a straight line well all I need is the gradient and I need a coordinate do I have a coordinate sure I have the center point so the equation of the radius is well y - y1 = m x - x 1 and then I'm going to have y - - 3 = 12 x - 2 do some algebra just to quickly get this in a good form then I'm going to move the three to the other side so we have y is equal to a half X and then minus 2 like that cool so now I have the equation of the radius and of course I have the equation of the T tangent so to work out the coordinates of this Center of this sorry of this coordinate here what I could do is simultaneous equations because both of these lines intersect at the point XY so if I equate them now so equating the equations I have- a half x - 2 is equal to 2x subtract 3 and then if we bring the X all to one side so I want to probably bring the X's to this side and the numbers to this side I think that makes the most amount of sense uh so that would be 2.5 so 5 over 2x which means X is just equal to 2 over 5 that's my x coordinate and then I can sub this into maybe the tangent equation might make more sense so then Y is = to 2 2 5 sub subtract three which is what - 11 five let's have a look see two 2 over five - 3 - 11 5 Perfection so now I have my coordinates okay wonderful now in terms of working out the distance I see a few students do something which is a bit interesting so you guys seem to try and memorize the equation right where it's like y y 2 - y1 2 + X2 - X1 2 yada yada yada I don't like memorizing stuff so instead I'm going to show you something really quick if I break this up into a right angle triangle that's just the hypotenuse how do you work out the hypotenuse of a right angle triangle you just use Pythagoras so it is actually just the Pythagorean theorem so difference in X is two and it goes to 2 over 5 so we can just work that out real quick and again it is the difference so it the actual sign doesn't matter it's going to be so I'll make it minus 1.6 but remember 1.6 and negative 1.6 is going to be the same thing because we Square it and it goes from - 3 to - 11 over 5 so we're just looking for the difference between those numbers so - 3 - - 11 over 5 uh - 45 cool so that is - 0.8 so the actual radius now and again we want it in serd form so what we're going to do is with the radius we're going to do square root of maybe I should leave it as a fraction then just to make it a bit easier I think that makes a lot of sense because that way we'll have it in a third form um but we could always check to see if it would give us the right answer so Square < TK uh 1.6 s again I'm ignoring the sign because it's going to be a distance anyway does that give me a third it does so that's fantastic never mind I don't need to do that so 1.6 squar I'll make that negative just for the working out so you guys know what we're doing but again with lengths we don't really care that much so it's going to be 4un 5 over 5 now that's a 10 mark question and I know it seems kind of long but for 10 marks that means you're going to spend about 10 minutes on it even when I was explaining that it's barely taking me 10 minutes so it's actually not that bad but can you see it really does bring together everything and also remember that when I read this question I didn't know what to do until I drew a diagram okay that's what you need to keep in mind even a maths Professor will not be able to just look at this and go oh yeah that's what we need to do you need to draw diagrams and that really does show The Examiner that you know why you're doing certain things it's really really important you can get marks for diagrams sometimes it counts as working out but that I will see you in the next question