 Welcome to the YouTube live session on problem solving on complex numbers for the main and J advanced So those who have already joined in the session, please type in in your name In the chat box so that I know who all are attending the session. Hello. Sorry. Good afternoon, everybody. Hello, Shruti Purvik started with the first problem Okay, M and X are two real numbers then find the real value of The expression e to the power 2 mi cot inverse x times you can treat this as this into X i plus 1 by X i minus 1 whole to the power M. Hello, Ritya. Hello, Ritik Sondarya. Good afternoon to you all So while we wait for others to join in you can start thinking on this problem So in this session, we are going to cover up various aspects of the topic we'll try to Take different questions from each and every sub topic of this chapter of Various levels J main and J advance both and if you to type in your response box whether it's option a Option is correct in this case Anyone who has made any progress on this So those who have those are joining in the session request you to type in your names in the chat box All right, guys. So as we solve this problem, I'll keep giving you some kind of clues or hints We'll start with the fact that let cot inverse X be theta Okay, so let a cot inverse X be theta and we'll first focus our attention on e to the power to I cot inverse X e to the power To I cot inverse X which is actually to e to the power I of 2 theta Okay So basically, this is nothing but the Euler form representation, right? We know that complex number Can be written in the Euler form as R e to the power I theta So if you compare this complex number e to the power I to theta with this form, it's actually one e to the power I theta Okay, and the polar form if you write the same complex number it becomes Cos theta plus I sin theta, right? So this is the polar form representation. This is the oil form representation So this becomes cos of 2 theta plus I sin of 2 theta Given that we have taken cot inverse X as theta it implies X is cot of theta Which was the implies tan of theta is going to be 1 upon X Now try to express try to express X number in terms of This complex number in terms of X Okay, so cos of 2 theta we know it's going to be 1 minus tan square theta by 1 plus tan square theta and Sin of 2 theta is going to be 2 tan theta by 1 minus tan square theta So using the fact that tan theta is 1 by X it becomes 1 minus 1 upon X squared by 1 plus 1 upon X squared plus I times 2 by X 1 minus 1 upon X right no problem so far Question so far any question in the simplification process Okay, so guys what we can do is we can simplify this further as X square minus 1 by X square plus 1 Plus 2 I X by X square Yeah, it's X square. Okay So that is going to be X square minus 1 plus 2 I X by X square plus 1, okay Now we all know that the numerator is going to be the expansion of X plus I the whole square The numerator is going to be the expansion of X plus I the whole square obviously because if you square this up You get X square plus I square which is minus 1 and plus 2 I X and the denominator if you see it is actually the Product of X minus I and X plus I right So X square plus 1 is basically The multiplication of X minus I or X plus I with its conjugate So one of the powers get cancelled over here Leaving me with X plus I by X minus I Okay now if you see the expression this desired over here is Almost of the same nature like this is just that I have X I instead of X and I have one in place of I So what I'll do now is I'll multiply the numerator and denominator with I So I'll multiply this with I and I'll multiply this with I Giving me I X minus 1 and I X plus 1 Okay, yeah, the power M is in which I'll take care of little later on So as of now what? to show you guys that e to the power to I caught inverse X is Actually I X minus 1 by I X plus 1 Okay, let's cross multiply this when you cross multiply this it becomes X I plus 1 by X I minus 1 Times e to the power to I caught inverse X is equal to 1 Okay, at the moment You raise both sides You raise both sides to the power of M to power of M Right there by giving you Thereby giving you e to the power to M. I caught inverse X Times X I plus 1 by X I minus 1 whole to the power M is equal to 1 to the power M right and Have they provided me any information about M? Yes, M is a real number. Okay, if M is the real number This is going to be one again That means my option number C will become correct. Is that fine? Any questions so far? Please feel free to type in any concerns or any questions on the chat box So now let us move on No, so here we have this question on your screen If Z 1 Z 2 Z 3 are three distinct complex numbers and a b c are three positive real numbers Such that a by mod z 2 minus z 3 equal to b by mod z 3 minus z 1 is Then what can we comment about The required a square by z 2 is a 3 plus e square by z 3 minus z 1 plus c square by z 1 minus z 2 Which two terms? Which two terms are you talking about? No Can't call them at each other their product is giving you a real number right Exactly, it should be mod of mod If Z Z conjugate is mod z squared then only we can say conjugates of each other Okay if the product number Real that that doesn't imply the two numbers are exactly conjugate It should also be equal to the mod of that number square, right side All right, so we have a response from pooric over here Okay, hold on the right answer because I want others also to reply So pooric we have I've noted down your answer. Let's see what what is the response of other people? Okay, Kushal says see anyone else shruti Ritwik Sundaria Sanjana sure sure take your time. All right, so Sai is also getting see wishes also Says should we start discussion of this problem? So we have this term a by mod z2 minus z3 equal to b by mod z3 minus z1 and so on So let's say I call this term to be lambda Now, we know these are all real terms. So lambda has to be a real number So a will become a lambda times mod z2 minus z3 Okay Which implies a square could be written as Z square mod z2 minus z3 square and we all know from the property that Mod of z square is nothing but z into z con Right, so I can write this a square term as a lambda square z2 minus deep times Z2 minus Correct Which is but minus z3 times z2 conjugate minus z3 conjugate right Now here if you see this expression This expression is basically obtained when you bring this on the denominator of a square. So a square by z2 minus z3 Will be equal to lambda square times z2 conjugate minus z3 conjugate right in a similar way we can say that B square by z3 minus z1 will be lambda square z3 conjugate minus z1 conjugate and Moreover c square by z1 minus z2 could also be written as Z square z1 conjugate minus z2 conjugate now. Would you like to change your answer? Anyone would you like to change your answer now? Yeah, now the options have changed for you guys Visist side Yeah, the network is slightly fluctuating guys, so you might experience some poor voice quality, but I'm sure the writing is visible clearly on the screen Okay, so when you add these three When you add these three that is summation of this quantity you would realize that you are getting lambda square Z1 conjugate minus z2 conjugate plus z2 conjugate minus z3 conjugate plus z3 conjugate minus z1 conjugate and Terms start cancelling each other out leaving you with the option as option a which is 0 Okay, so here the right option is your option number a is that fine guys So in this property, I just tested you upon your basic understanding of the conjugate property Without wasting much time. Let's move on to the next problem, which is problem number three So here they have asked you the point of intersection of the curves whose argument given by argument z minus 3i is equal to 3 pi by 4 Okay, so we have curve one whose Equation is argument z minus 3i is equal to 3 pi by 4 and we have another curve c2 Which is given by argument 2 z plus 1 minus 2i is equal to pi by 4 So they have asked the point of intersection of these two curves if you're done, please type in in So hint is try to solve this problem By sketching these two curves on the argon plane So basically this is testing your geometrical idea or geometrical clarity about The terms like argument of or minus some other complex number, etc. So kushal is saying option a Okay What about the others? Take your time. No need to rush accuracy is more important Hope you have started taking the J. E. Mock test and you are working out on the strategy and Also the mistakes that you are making on a regular basis. I the first welcome So I really says option B Precious poor week. Okay. It's first. Let us understand over here when you say mod of z1 minus z2, what does it mean? It means the distance between It means the distance between z1 and z2 Distance between the complex number Z1 and z2 Right. This is clear to all of you when you see the argument of Z1 minus z2 Z1 minus z2 is theta. What does it mean? It means that If you make an argon plane If you make an argon plane and there is a complex number z1 over here and there is a common number Okay Then argument of z1 minus z2 is make a vector make a vector from z2 to z1 Right the direction of this vector should be from z2 to z1 right and Extend this Extend this vector backwards So wherever it cuts the or whatever is the angle that it makes with the real Z axis That would be the argument of this complex number right so if I say what is the argument of What is the argument of? Z2 minus z1 Right, then how would he do it? Then you make the arrow in the opposite direction. So I'm showing it in blue color It will be an opposite direction. So when you extend this backwards Okay, this would become your argument of This would become your argument of Z2 minus z1 getting this point right at this fight and this theta would be the argument of This theta would be the argument of Z1 minus z2 this clarity has to be there because I see a lot of confusion regarding this Hope nobody has any question with respect to what is the meaning of or how do you Isn't on the argon plane the argument of z1 minus z2 Right now in this case, let us see what is there for us again. Let me draw an argon plane now the first curve says that Argument of z minus 3i is 3 pi by 4 and z is basically a complex number, which is not known to us How should Z move so that its argument z minus 3i is always 3 pi by 4 The first of all Where is your 3i? Where is your 3i located 3i is located on the z axis on the imaginary z axis as a 0.0 comma 3. Okay, of course If z lies in its argument will always be lesser than pi by 2 correct So it cannot lie in this direction So this is Not possible Can I say Z will lie in this direction? Correct if that lies in this direction then I can say if I extend this backwards So if I make a vector quantity like this if I make a Vector like this assuming that here lies my Z Then this of course if you extend it backward you can say yes, this can make an angle of 3 pi by 4 Yeah, Nitya a lot of things can be done on this But what I'm focusing on is how you understand these things geometrically Because sometimes geometrical solution is much more easier than doing it in a mathematical way Now guys, just remember one small thing that when you are making such Geometrical representation Z cannot become 0 comma 3 so there will be a hole over here. So please note there will be a hole over here So it can take any Value in this line. So this can extend indefinitely. This can extend indefinitely so curve C1 is a line which is going to be a Originating from 0 comma 3 and going in this direction as shown to you correct in a similar manner if I focus on the other curve the other curve is argument of 2z plus I 2z plus 1 minus 2i so correct. So if I write it as 2 Z plus half Minus I can be written. Yes or no This is nothing but argument of 2 plus argument of Z minus minus half plus I Correct, which is actually this is actually 0 because argument of any positive real number is 0 So now it is given that this argument is equal to pi by 4 Again, where is minus half plus I located? Half plus I would be located over so X would be negative half and Y is I So this is your minus half comma one Now just saying argument of Z minus. This is pi by 4. So can I say Z will be in this direction? so Z would be in this direction Correct Z can be located anywhere in this direction and this angle is going to be pi by 4 This angle is going to be pi by 4 correct now you would realize that such Assess through minus one com minus half comma one and inclined at an angle of pi by 4 Will never ever meet this red line So black line and red line are never going to meet are they ever going to meet tell me you can individually Equations if you want so equation of the black line is going to be this equation of the black line is going to be Y minus one is equal to One times X plus half So it's Y is equal to X So the moment you put X equal to zero your Y is equal to three by two So it will cut here cut at three by two here zero comma three by two So they will never meet So these two lines will never meet If they never meet or if they never intersect they cannot be any point of intersection So your option number D becomes correct over here So your option number D becomes correct over here. They have no point of intersection. Is that clear guys? Any question so far on this? Another method is where you can take your Z to be X plus I Y and try to simplify it by using your idea of argument and get to different equations and see whether and Plot them on the argon plane or you can say real ax real Cartesian plane and see whether they intersect or not However, when you're going to do that, you are going to realize you're going to get these two curves these two lines itself Okay, so don't waste time getting into simplification of those equations if You understand the concept how to deal with it geometrically you can save a lot of your time So now let us move on to the fourth question over here All of you please read this question carefully Let a1 a2 till a n be the vertices of a regular polygon of n sites inscribed in a circle of Radius unity So two words to be taken care of It's the regular polygon of n sites Inscribed in a circle of radius one. So just for your sake of reference, I'll make a Argan plane diagram on this so basically, it's it's a Polygon like this Inside a regular polygon. So I'm not drawing the entire polygon. I've just drawn a part of it We have been given a and small a and small b and we have to find the ratio of small a and small b So I think this is a integer single integer type question This is a beautiful question. This will test you on your understanding of and it Roots of unity. How did I link it to nth suits of unity? Because as you would all know that The n nth suits of unity if you plot them on the Argan plane. They are located On the vertices of a regular polygon inscribed in a circle of radius unity That's why I told you these terms are very very important regular polygon of n sites Inscribed in a circle of radius unity the moment you read this word this nth roots of unity can should come in your mind Right, that means the vertices coordinates or you can say the complex number which represent the vertices of this n-sided regular polygon N-sided regular polygon would be nothing but your n nth roots of unity there would be nothing but your n and nth roots of unity Read this as a hint and see whether you can make any progress on this. How many of you have very less understanding about nth Unity so basically the equation z to the power n equal to 1 Okay, and I'll just briefly tell you so that people who are not aware of this concept can start at least working on this problem the nth roots of unity the nth roots of unity is basically cos 2k pi plus i sin 2k pi by n where where k takes values from 0 1 2 all the way till n minus 1 okay Where k takes in values all the way from 0 to n minus 1 So ideally your first route should have been at at the origin. Okay, so here would be your first route So a1 should have been over here. This is your a1. This is your a2. This is your a3 This is your a4 etc. So for nth roots of unity. You will always start with your On 1 commas easy k as 0 you'll get your first route as 1 comma as z equal to 1 Which is nothing but 1 comma 0 point So this is a 1 comma 0 point a very important thing about cube root of unity is that If you put k as 0 you will get 1 You will get z as 1 If you put k as 1 you will get z as cos 2 pi by n plus i sin 2 pi by n Which we normally call as alpha. Let's say this is alpha So when you put k as 2 you will get z as cos 4 pi by n plus i sin 4 pi by n Which automatically becomes alpha square, right? This is in relation to which theorem If this is alpha Then this is alpha square is stated by which theorem Please type it in the chat box Is that my audible to you? Right, that's a demo of this theorem demo of this theorem very important thing So guys just keep make keep making note of these things whatever you need to revisit again So nth cube nth roots of unity is one of the very important concept which you all need to revisit again Okay, so if you keep on plotting, let's say k is equal to 3 then you get cos 6 pi by n plus i sin 6 pi by n which is that nothing but alpha cube, right? So it keeps on going till you reach n minus 1 Where you will get cos 2 n minus 1 pi by n plus i sin 2 n minus 1 pi by n and this can be written as alpha to the power n minus 1 Okay, so you can see that The roots are actually in our in geometric progression, right? So the roots will look like this the roots will be 1 alpha alpha square alpha cube Till alpha to the power n minus 1 A very important property that these roots will satisfy is that their sum will be zero Okay, their sum will be Equal to zero always because this equation z to the power n minus 1 this equation doesn't contain a z to the power n minus 1 term, right? So there is no z to the power n minus 1 term. So the sum of the root has to be zero Okay, and the product of the roots Alpha alpha square Till alpha to the power n minus 1 is given as minus 1 to the power n minus 1. So just remember these two very very useful results So see whether you can utilize this concept in solving this problem So guys, please feel free to type in your concerns or queries. We still haven't found a and b yet We just talked about the the what are the nth roots of unity and what is the important property that we need to keep in our mind Okay, should I help you with this? Should I start solving this? or anybody is in the middle of his Solving so let's let's start working out on this. So the expression a the expression a over here is uh summation of mod Uh a1 You can say a1 a2 a square a1 a3 square and all you can say a1 ar square. Okay, and r is from Yeah, sure. Sure. Take some time r is from uh 2 to n isn't it r is going from 2 to n mod a1 ar means the distance between a1 and ar So let's say this distance This is what they are referring to as the distance a1 ar Okay, so let's let's talk about it because we can't spend a lot of time on this because we have other problems as well a1 ar is basically nothing but mod of mod of The distance between a1 and ar Right a1 is at 1 ar will be nothing but alpha to the power r Correct where alpha I've already told you alpha is going to be cos of 2 You can say 2 r pi by n Plus i sine 2 r pi by n. This is your alpha to the power r I've already explained you in this case alpha alpha square alpha q n so on now If we use the polar form over here It becomes cos 2 r pi by n plus i sine 2 r pi by n mod Which is nothing but mod of 1 minus cos 2 r pi by n plus i sine 2 r pi by n Mod of this Let's check. Let's check for it. It will be in terms of r only. Let's check it so this is going to be a under root of This is going to be under root of 1 minus cos 2 r pi by n whole square Plus sine 2 r pi by n whole square right So if i'm not wrong, it'll give me 2 minus 2 cos 2 r pi by n under root Yes, I know that the square of a 1 a r Square of this will become 2 minus 2 cos 2 r pi by n Now what do I need to find? I need to find your a term Which is nothing but summation of this from Summation of this term from r equal to is there any questions? Now, how do I find this out? Of course, I can break it up as two summations One is from summation of 2 from r equal to 2 to n And other would be minus 2 summation From r equal to 2 to n of cos 2 r pi by n Right, so that will give me two times What how many terms are there from n? from 2 to n From 2 to n if you count, how many terms will you get? Type it in the chat box Will it be n minus 2? Will it be n minus 1 or will it be n? Right, that would be n minus 1 correct, Vishisht And do you realize that guys? I don't know how many of you realize this This term is actually the real part of alpha to the power r So can I say when I'm summing this up, it is going to give me two times real part of Real part of 2 alpha to the power 3 and so on Till alpha to the power n minus 1 A little while ago we discussed I think let me just Scroll up the screen Not very far What is 1 plus alpha plus alpha square plus alpha cube All the way till alpha to the power n minus 1 that is equal to 0 Right, so This can we say Alpha alpha square all the way till alpha to the power n minus 1 is going to be minus of 1 Right, so this result would give me 2 times n minus 1 minus 2 the real part of it would be minus of 1 And that leaves me with only 2 n. So I finally got my a So a is equal to 2 n A is equal to 2 n Is that clear? Any problem in understanding how I evaluated that the number thus the expression for a Is it clear clr? Please type in clr if it is clear to you so many concepts Revisited in this problem. We came to know about the positioning of the nth roots We came to know the property of it Okay. Now guys, we have still half the work left. We have to find your expression b also Okay, so b is given by Product of all these things b is given as the product of all these things Okay, so let's work on b now so b is your mod of 1 minus or you can say product of mod of 1 minus alpha to the power r From r is equal to 2 to n so r is equal to 2 to n Let me just write a 1 a r a 1 a r from r equal to 2 to n Yeah, so a 1 a r we had already discussed this previous attempt of finding a Right. So a 1 a r A 1 a r is going to be this term Okay But anyways, I'm not going to use this while solving this problem. I'm going to use a different approach over here This problem you are actually finding out The distance between 1 minus alpha Multiplying it with 1 minus alpha square Multiplying it with 1 minus alpha cube And so on till you reach 1 minus alpha to the power of Alpha to the power of n minus 1. This is what we are actually doing So we are finding the product. Let me call it as b here So we are finding the product of the distance of 1 minus alpha 1 minus alpha square and so on It is the same thing as if I'm trying to find out the mod of Product of 1 minus alpha 1 minus alpha square 1 minus alpha cube and so on till 1 minus alpha to the power n minus 1 Okay Now guys, uh, please be informed that 1 alpha Since 1 alpha alpha square etc to the power alpha minus 1 are roots of Are roots of z to the power n minus It implies minus 1 can be factorized as z minus 1 z minus alpha z minus alpha square All the way till z minus alpha to the power n minus 1 Right Right. So if this is the root If these are the roots of this equation We can always factorize this expression We can always factorize this expression like this Okay Now bring this z minus 1 down over here So z to the power n minus 1 by z minus 1 is z minus alpha z minus alpha square All the way till z minus alpha to the power n minus 1 Now if you look at this expression carefully You need to put your z as 1, right? You need to put your z as 1 Right when you do that on the right hand side you start getting 1 minus alpha 1 minus alpha square and so on till 1 minus alpha to the power n minus 1 Right Whereas on the left hand side you will have to take the limiting case You will have to take a limiting case the limiting case requirement is because it is 0 by 0 form if you put z as 1 So you take the limit z 10 to 1 Right now this is a standard Limit standard algebraic limit which you all have studied that is n a to the power n minus 1 Right, so which is going to be n actually So here comes my value of b for you Okay So summarizing a was your 2n This was your a This was your b Right So what will be a by b a by b will be 2n by n and therefore your answer is going to be 2 Is that clear any question? Yes, it was slightly lengthy question. But if you are aware of the concepts of The nth roots of unity you should be able to solve this within five minutes Any that you want to ask over here? Please feel free to type in in the chat box All to the next question Which is your question number five Which is your question number five? This should be an easy question guys. I want everybody to answer in this case Yeah, sure. Nikhil, uh, what I I'm just going back to the once again So what I did is I know one alpha alpha square till alpha to the power n minus one are roots of the equation Z to the power n minus one equal to zero. What is the meaning of roots roots means? Uh z minus one z alpha, etc will be the factors of this particular expression Right now all I did is I substituted as z as one On the both left and the right side of the equation On the right side. I ended up getting the desired result Okay, whatever I wanted But on the left hand side, I have to take a limiting case because when you put z as one directly over here, it will become zero by zero form So I evaluated the limit by using our Remember this z remember this algebraic limit which you had learnt in class 11 This is n a to the power n minus one right in a similar way a is going to be one over here The role of x is being played by z So it becomes n One to the power n minus one that's equal to a really real number its mod will be n itself So a becomes a b becomes n a was already 2 and we discussed it and hence the ratio becomes 2 Is that fine nickel? This is a pretty simple question. I think you should be able to answer it pretty fast If you're done, please type in your response in the chat box So I say option b Okay What about the others? Guys the hint for this is you have to use your understanding of the rotation theorem right Also called as the coney method think in lines of the coney method if uh If you have forgotten this rotation formula, let me just quickly remind you of what is the rotation formula rotation formula says that if you have three complex numbers which are The affixes of this particular structure which you see on your screen Where this angle is theta angle Then the relationship between z 1 z 2 z 3 and theta will be given by the expression z 3 minus z 1 by z 2 minus z 1 is equal to mod of z 3 minus z 1 by z 2 minus z 1 e to the power i of theta This is called the coney method For the rotation theorem on complex numbers Okay, very very important Please remember this It can be used at so many places. I want to ask you Uh, we have a curve traced by z, right? What is this curve? I mean Does this equation give you any idea about what curve z is tracing? If anybody knows what curve z is tracing, please type it in the chat box Is it an ellipse? Is it a circle? Is it a parabola? Is it a straight line? What is it? What curve is it tracing? Okay, so Purvik says pair of straight lines Sai, Nishal, Kapas, Nitya Kushal is saying it's tracing a circle I'm not saying right and wrong to anybody Vishesh is saying ellipse Okay So taking a clue from a coney method over here Can I just write this expression? Correct me if I'm wrong Can I write z plus i by z minus i like this? Can I just do this thing? Uh i plus z By minus i plus z. Okay. I've just reversed the position of z plus i and z minus i in minimum rate and denominator Which is nothing like which is nothing but saying z Okay, or you multiply both sides with the minus sign as well. You can do that So minus i minus z is equal to is equal to i minus z Okay Now why I did all these things is I wanted it to resemble this expression I wanted it to resemble this expression. Okay. See the resemblance between these two expressions Who's playing the role of z? Who's playing the role of z three when you compare these two expressions You'll say z three is nothing but your minus i Who's playing the role of z one z one role is being played by z And who's playing the role of z Z two role is being played by Z two role is being played by i right Yes or no Now This coney method says This is the argument of This is our argument of You have written This is the modulus of right So indirectly what I want to say is that this is a complex number whose argument is given to you Isn't it? Yes So this theta becomes the argument of Of This entire I have been given this angle theta to be pi by four in the question Okay, so let us The position of at least z three and that one so z three will be somewhere over here Z one will be somewhat somewhat over here, right That three is here and z two is here, okay And z one is somewhere free to move Right, so this is free to move While these two are fixed. This is fixed and this is fixed but z one is moving in such a way that The angle it makes Let's say I take this position over here This is your z one. It moves in such a way that the angle is always pi by four This angle is always pi by four First it is making a positive angle So I this is minus I This is making a positive angle Pi by four positive angle clockwise angle So this is the scenario Right guys had I said it is a negative angle. How would you make that angle? How do you had this angle be negative pi by four? How do you make this angle? Always remember This angle sign Is dependent upon its clockwise or anticlockwise sense theta is positive if you take it in an anticlockwise sense somewhat like this had your Z three z one z two like this Then the same coney method formula would have been written like z two minus z one by z three minus z one equal to mod z two minus z one z three minus z one e to the power i minus i theta So the way I remember this and I think last year also I told you how to remember this You can make this angle with any arrow or with any direction you want to right for example, if you make this and clockwise like this Then always start writing these two complex number first z two minus z one, which is your numerator And then these two complex numbers Which is in the denominator Same thing modulus e to the power i negative theta negative theta because it is clockwise I'm getting this point. So both are coney method, but one taken Another taken in an anti clock and you would realize that if you reciprocate it it will become the same right now So if my pi by four is a positive angle over here I always have to make it in a anti clockwise sense Make sure you have written this Minus z first Have you written this minus z? Yes, I can see you have written Z three minus z first. So it is a correct representation so This you can say that this point z is moving in such a way that this angle will always be pi by four That means it can move in this way also It can move in this way also So all these angles are going to be pi by four pi by four pi by four because Along a It is circle Who's Is the Connecting So the locus of it would be a part of a circle now Once it will not be a complete circle. It will not go in the other side. This part will not be there Okay, so this part will not be there This part will not be done yeah Now you tell me you have to find the length of this arc this arc length we require Right Can I know the angle? Can I know the angle subtended by this arc at the center of the circle? So basically i'm redrawing it by using my Tool of a circle over here So let's Okay Now this is your arc. Let's say connecting This is your arc connecting i and minus i let's say This is i point. This is minus i point And you don't have this part You don't Right and let's say this is my center and I want to know what is this angle I want to know what is this What will you say? Wow, please reply on the chat box. What is theta? First given that this angle was Given that this angle was pi by four This angle was pi by four What is this angle going to be and then what is theta reply on the screen? Please reply on the chat box. Yeah, this is going to be pi by two, but what is going to theta? Theta is going to be correct theta is going to be three pi by two So your theta angle is going to be three pi by two Right circle, if you know the angle Right, we can always find the length of the arc. So arc length is going to be r theta Right, but the problem is we don't know our right Circle Right for that you don't have to break your head much because If you drop a perpendicular If you drop a perpendicular from The center on this, you know, this is going to be 90 degree and this is going to be 45 degree So 45 degree 45 degree And this is going to be uh One unit Okay Yes So if this is one, this is also one. So this will be root of two So radius will be root of two in this case Just a simple geometry. So my answer would be r into theta Right, so Will be three root two pi by two. Do we have an option like this? Three root two pi by two. Of course, we have an option option number c is correct. Is it clear guys? any question Let me know Okay so again this Concept tells us that we need to be very very good in our kony rotation formula And especially when the what side of The circle is uh, no required in that locus is something which we need to watch out for Because if you take a wrong arc, your answer will be different the trick Right, so always The arrow on the numerator expression And tail at the denominator expression And see what is the angle sign If your angle sign that you get matches with the angle sign that is given in the question That means you have taken the right side else you switch your side Is that clear? So with this we move on to the next question Which is So here we have a question z 1 z 2 z 3 are the vertices of the triangle abc and z naught is the circumscent We have to find this expression is equal to 0 1 minus 1 or none of these Okay, so i've just done the diagram for your reference. This is a b c Circumcenter is z naught This is z 1 This is z 2 This is z 3 See Does these expressions remind you of something guys if you look at these two expressions take a clue from the question Does this remind you of something? If I just switch it and write let's say I write it as z 1 minus z naught by z 3 minus z naught Does it remind you of something? And here also if I switch it and write z 2 minus z naught by z 3 minus z naught Does it remind you of something if yes Please write the expression for it you your Come to what's going on. So, what does this first expression remind you guys? Please Doesn't it remind you of kony method? right So this means you're writing kony at z naught Right, so this means you're writing kony at z It's for this I this angle will be 2 b yes or no And see That is the numerator term. So your arrow should be pointing like this so in other words you are writing this as a kony formula which is going to be mod z 1 minus z naught by mod z 3 minus z naught e to the power i Now this is an anti clockwise direction. So you'll write as 2 b. Is that clear? And similarly for this z to my writing kony at You're writing kony at this position Isn't it again head of the arrow should be the one connecting z naught and z 2. So this is the head of the arrow So when you write kony over here, what you write mod z 2 minus z naught by mod z 3 minus z naught z naught equal to Into e to the power i This angle will be 2 a but since you're taking it at the clock, you'll write minus 2 a Any question? So for this type something on your screen Is this equation 1 and 2 clear in your mind? Yes, no, maybe Okay Great. So now Can I say that the distance of can I say z minus z naught Distance would be same as z 3 minus z naught Would be the same as z 2 minus z naught and each will be equal to the r which is the circum circle radius Right, we all know that Circum circle the point which is equidistant from All the three vertices of the triangle, right? So can I say 1 reduces to this z 1 minus z naught by z 3 minus z naught equal to e to the power i 2 b Because this expression will cancel out and become 1 correct Which is nothing but cos 2 b plus i sin 2 b And similarly your expression 2 becomes z 2 minus z naught by z 3 minus z naught equal to e to the power This which is actually cos 2 a minus i sin 2 a right now Having got this look at the expression the expression says I have to evaluate this So let's plug in the values So cos 2 b Plus i sin 2 b Times sin 2 a I'm going to take sin 2 c as lcm because sin 2 c is present in the Denominator part of Of them Okay, right to cos 2 a minus i sin A over its b Yeah, I sin 2 a into sin of 2 b if you open this up the real part terms would be cos 2 b sin 2 a plus sin 2 b cos 2 a Okay, imaginary terms would be Sin 2 b sin 2 a plus sorry minus Minus sin 2 b Sin 2 a Hold divided by Sin 2 c Now this and this gets cancelled Right, so I'll be only left Okay Yeah, now since we're dealing with a triangle we know that 2 a plus 2 b plus 2 c is going to be 360 degrees So 2 a plus 2 b is going to be 360 degrees minus 2 c So it's going to be sin of 360 degree minus 2 c by sin 2 c Which is going to be minus sin 2 c by sin 2 c And sin 2 c sin 2 c can be cancelled out leaving you with the answer as minus 1 Is that fine guys? Is that fine? So yes Time here Purvik you're correct. The answer for this problem is now going to be option c which is Minus 1 Any question so far? Again, this was a good problem where you could Use your concept of geometry as well as the Korni-Rodin formula in tandem to solve this problem Right excellent problem guys Moving on to the next problem We are almost halfway through So f of x and g of x are two polynomials This says that h of x is x times f of x cube plus x square times g of x to the power 6 Which is divisible by x square plus x plus 1 Then which of the following option is correct? This can be more than one option correct question. Okay multiple options may be correct in this case So, please be very very careful. Do not miss out any option while solving it Of course it does Yeah, there may be several occasions where you Can't even figure out whether complex number is applied to that concept So now welcome to the world of problem solving. That's where you have to think in all dimensions right Now there's no difference between real and complex and all all concepts can be used within one another This is a beautiful question. I mean I mean, it's very simple as well So I am again. I'm repeating it's more than one options are correct question. So please type in your response In this does this term remind you of something What clue do you get from this? If I ask you What are the roots? Right omega and omega square absolutely correct. So basically x square plus x plus 1 can be factorized as x minus omega and x minus omega square Right where omega and omega square are your complex cube roots of unity Correct. Everybody knows about omega and omega square over here Okay, so those who don't know let me tell you very quickly over here. Omega is basically a complex cube root of unity complex cube root of Unity its value is minus half plus i root 3 by 2 Okay, this is your omega anyways guys, uh So if you're seeing that This is divisible by this correct means your h of x is uh You know some function. Let's say a k of x of x square plus x plus 1 right In other words, it can be written as some function k of x times Don't treat it as k into x. It's a function. Okay x minus omega x minus omega square Right Doesn't it imply that putting omega in place of x will give you zero and putting omega square will also give you zero Okay, so with this as a hint Take this as a hint And tell me solve this and tell me which options are going to be correct Very simple problem guys very very We are more than one option correct. You should never take a guess Others Shruti Sanjana Ritya S please type this once Let it be long. No worries Okay A and c Okay All right, so let's see what happens When you say at omega is zero, which means you're saying, uh Omega f of omega cube Plus omega square g of omega to the past six is zero Which means omega f one plus omega square g one is equal to zero Okay, let me call this as one Similarly, when you put omega square, you're saying omega square f of omega to the power of six Plus omega to the power four g of omega to the power of 12 Is equal to zero Is equal to Zero right Which means omega square f one plus omega f one is equal to zero Sorry g one is equal to zero. Okay Now treat this as if we have two simultaneous equations in x and y and we have to The value of that Okay, so let us try to element first find f of one By eliminating g so let us multiply two with omega So let's multiply two with omega so i cube f of one Plus omega square g of one equal to zero Now subtract these two equations subtract it you get Omega minus omega cube f of one Equal to zero Now we know that Omega cannot be one Right because omega is a complex cube root of infinity. So it implies f of one has to be zero Which implies g of one will also be zero because if you put zero over here g of one will also be zero Right, that means f of one and g of one are definitely the same And they can be negative of each other also because zero can be negative of zero also, correct So c cannot be there and d cannot be there. So c and d are ruled out So your options a and b become correct in this way Not a and c a and b I think you guys got carried away with the answer given by the first person Okay, so be very very careful while marking such options. I have seen people losing 8 to 12 marks because by mistake they have marked some other options Right, so don't do all these things Do your basics correctly first Of course zero is minus of zero, right? Isn't it correct? Sorry, isn't zero my negative of zero Zero is the only number which is negative of itself. All right So now moving on to the next question, which is your question number nine for the day after this will take a small break And do the rest of the six questions Very simple question if the roots of this equation are of opposite Then the point represented by the complex number a plus i b is located inside outside lies on a straight line, none of these This option correct guys. So you should be very fast in solving this. I should have got a response by now This is a typical j main problem. It is not j advance. It's a it's a j main problem Okay. Purvik has given an answer Sai says b Nitya also says b kushal b Let's see kushal. I'll not disclose the answer right now. Let me wait for responses from other people aditi shruti tapas which is rithvik nikhil k rithvik also says b All right, guys, uh, this is a super simple question. I was expecting the answer to come from everybody Those who have said b. Yeah, you are correct So answer is b So when are the roots of opposite sign? We know that the product of the root is c by a right in a quadratic The product of the roots alpha beta is equal to c by a and if it if one of the roots has to be Uh, positive another has to be negative. This has to be a negative quantity. Correct So c uh in this case Will be minus b square a square b square plus four by b square this should be negative That implies this is always positive. There's always greater than equal to zero So the only possibility is minus a square minus b square plus four should be negative Correct, that means a square plus b square should be greater than four That means mod z square Is greater than four, which means mod z will be greater than two Now it clearly implies that if mod of a complex number is greater than two That means the distance from the origin should be greater than two. That means it should be outside a circle It should be outside the circle Whose center is at origin and radius is two So it will be outside the circle center at origin and radius at two. So option b becomes direct option Okay So guys now, let's uh, take a quick break Okay, let's have a break now and uh, we'll meet after around seven to eight minutes Okay, so let's resume our self at Right now it's 5 15. Let's resume at 5 23 p.m. Okay. Let's resume at 5 23 p.m. All right guys Welcome back after the break The next question in front of you which is your question number 10 for the day Given that I've discussed so many things about the Roots of this being easy question for you guys So the question reads if n be an odd positive integer. So remember n is an odd positive integer And uh one alpha one alpha two till alpha n minus one are nth roots of unity Then they're asking you the value of the expression Two plus alpha one two plus alpha two Two plus alpha three and so on till Two plus alpha n minus one. Okay. So purvic has already given the answer See, let's see what others have to say about this Anybody else please feel free to type in your response in the chat box again Try to recall that Alpha one, sorry one alpha one alpha two etc Till alpha n minus one are roots of by the way, they are Subscript one two etc can also be the powers Okay, they are the roots of this equation That means you can always write this as Nikhil was asking this question the other time. So we can Always factor up this like this right Now if you see I have two plus two plus two plus every time, correct Okay, so first thing what I'll do. I'll bring this z minus one down over here So z to the power n minus one by z minus one is z minus alpha one z minus alpha two Z minus alpha three and so on till z minus alpha n minus one okay Now the only option that appears to me right over here is that if I put z as two Sorry minus two Right put z as minus two. See what happens On the left hand side, you have minus two to the power n minus one by minus two minus one Right hand side, you'll get minus two minus alpha one minus two minus alpha two And so on till minus two minus alpha and minus one Okay, so you can play with this particular factor and we want to Now, uh, since there are Since n is an odd number Okay, since n is an odd number as given to me in the question n is an odd number Tell me from Alpha one to alpha n minus one How many terms would be there? Will it be even or will it be odd? So will I have odd number of terms or will I have even number of terms? odd or even If I go from alpha one till alpha n minus one and n is odd This n is odd I will even number of terms or odd number of terms Let's say n is five So from alpha one to alpha four This is even right Right So this will be even number of terms And if it is even number of terms, I can see that the signs will all be adjusted on the right hand side So you can Write this as two plus alpha one two plus alpha two and all the way plus till two plus alpha n minus one Okay On the right hand side, you realize that since n is an odd number This will become minus two to the power minus two into two to the power Right And similarly minus one you copy and this is minus three Right now get rid of the negative sign if you get rid of the negative sign You will end up getting two to the power n plus one by three Which is your option number c is that clear? You could have also taken A very simple example in this case where you have taken n as three So when you take n as three basically what you get You get One omega and omega square as the the cubes. So you are asked two plus one Sorry two plus omega And two two plus omega square. So when you multiply it you get four Then you get omega cube Then you get two times omega plus omega square Which is going to be Which is going to be five Minus Which is three. I have done the multiplication correctly two mega square minus one four plus omega cube Right. So here if you put n as three you get two to the power three plus one by three which is going to be nine by three And hence it satisfies the given condition So I could have even used my special case to solve this problem Right forget That even some special cases can be used to break such problems Right easy problem Now let's go on to the next problem, which is your problem number 11 for the day So read this question very very carefully DJ type question So basically what they're trying to say is that r1 Is the maximum distance of any point on this curve from the origin so r1 is the maximum distance of any point On the curve given to you from the origin And r2 is the minimum distance They're asking you the r1 plus r2 In such cases, I always recommend use z as x plus i y and try to get the curve first Then you'll get a better idea about Now how is this curve positioned? What can be the maximum and minimum distances? Any idea about this? Okay, so let's first simplify this expression Uh z z conjugate would be mod z square right And then we have three i This can be written as z plus z conjugate times z minus z conjugate minus 6 e equal to 0 So this is going to be x square plus y square minus i z plus z conjugate will be 2x z 2 i y minus 6 equal to 0 Okay, so overall we have 10 x square plus y square And i square is minus 1 so we'll have 6 into 12 x y minus 6 equal to 0 5 x square plus y square Plus 6 x y minus 3 equal to 0 Minus 3 equal to 0. Is that fine guys? absolutely this curve Right, so Purvik is correct. This is basic ellipse Major and minor accesses are not parallel to the x and the y axis. Let me show you this curve actually So let me just show you this on the geojabra tool So it's 5 x square plus 5 y square minus 6 x y sorry plus 6 x y Plus 6 x y minus 3 equal to 0 Okay, so as you can see on your screen It's the case of an oblique ellipse Okay However, it is centered at origin, right? It is centered at origin How do you find the center of an oblique ellipse? So if you want to find the center of this assume s to be 5 x square 5 y square plus 6 x y minus 3 Do dou s by dou x equal to 0 and dou s by dou y equal to 0 and simultaneously solve these two So when you do dou s by dou s dou s by dou x equal to 0 you get a 10x uh plus 6 y equal to 0 and when you do dou s by dou y you get 10 y plus 6 x equal to 0 Clearly suggesting that x and y both will be 0 in this case So this is an ellipse whose center will be at center will be at origin as we can also see from the geojabra diagram right Now when you know the center is at origin you can still take any point on this ellipse as a parametric point r cos theta or you can say Let r cos theta comma r sine theta be any point on this curve Okay, so instead of taking a generic point. I am taking r cos theta and r sine theta as Any point on the ellipse Okay, now this r and theta both are variables in this case right, so both r and Theta are variables So basically I've chosen a polar coordinate Okay Now when we substitute it over here Why I why I've taken it as a polar coordinate because the question talks about distance from the origin Distance from the origin and there my r will be helpful to me So what I'll do is I'll substitute X as r cos theta and y as r sine theta here. So I will get this will become r square This will become six r square sine theta cos theta minus three equal to zero, right Which means I can always write r square minus three equal to zero That means r square could be written as three divided by five plus I can write six sine theta cos theta as three sine two theta Okay, so r square is this Now if I ask you What is the maximum value that r square can take? What is the maximum value of r square? What is the maximum value of r square? You'll say the maximum value of r square will be when sine of two theta becomes minus one, isn't it? So that becomes three by two correct And what is the minimum value of r square? You'll say it is when sine two theta becomes one That's three by eight Right so far so good Any concern in this? Now r square You are saying it lies between three by eight to three by two Correct that means Mod r which is actually r is always positive. So I don't need to write mod r but still R will always be between root three by root eight root three by root two Correct that means the minimum value of r is r min is going to be root three by two root two And maximum value is going to be root three by root two This is what we call as r one in this problem and r two in this problem Right So let's add them So r one plus r two will be root three by root two into Half plus one which is going to be three by two. So it's three root three by two root two Right should have been an integer problem, but it has given some different answers. Okay never mind So this becomes our answer for this case Now see the depths to which they are Questioning you they have gone not only for you to realize an oblique parabola Then this was a very critical point. This was a very critical thing that you should all be You know taking up And then from here you have to find the max minimum value Which is again, you know taking you to the trigonometric equation part So one question is basically covering up so many concepts simultaneously. Okay Good. So with this we can move on to the next question, which is question number 12 for the day Again, this should be an easy question guys This is a complex number equation from where you have to get your complex number again The suggestion would be to use a x z as x plus i y in this case So I'll suggest using z as x plus i y because we are Very used to recognizing curves in their Cartesian form rather than in their complex number form or polar form Any responses guys so Concepts which are based on locus or application of geometry in complex number and it roots of unity rotation formula De Moivre's theorem Basic understanding of modulus and argument these are of very very importance when you're dealing with complex number chapters So please revisit these Topics and solve as many problems as you can What is the equation of a rectangular hyperbola rectangular hyperbola can be of various shapes? So example, it can be of the form x minus It can be of the form x y equal to c square Right It could be of the form x minus alpha square minus y minus beta square is equal to c square form So depending upon which form you are trying to find out it can be of these forms Here also it can become x minus alpha y minus beta equal to c square or square of something or some number Yes, guys done shuti rhetoric kushal mere Any response for you guys Option a okay option b b one second One second is not an option shuti aditi sanjana aditi Rectangular hyperbola, okay from the girl side anybody sanjana Aditi Sai mere you have changed your answer for the third time now All right guys So what I'll do is I'll also take the simultaneously the polar form of this curve Okay, so x plus i y and I will also take z as r cos theta plus i sin theta, right? Uh, because you see half the equation is basically using the x plus i y form half the equation is basically using the The the polar form, right? So it's better to take both of them. However, there is a direct correlation. You can say x is r cos theta And y is r sin theta All right now mod z minus 2 minus i will be x minus 2 times y minus 1 square under root right And mod z is r actually And this is sine pi by 4 minus theta which is sine pi by 4. Let this be theta So sine pi by 4 cos theta minus cos pi by 4 sine theta Okay So it becomes a x minus 2 whole square plus y minus 2 sorry y minus 1 whole square under root equal to r If you expand this you're going to get 1 by root 2 cos theta minus 1 by root 2 sine theta Okay Which is nothing but 1 by root 2 r cos theta minus r sine theta Right So this is what we get. So finally, this is what we get Now guys, uh, since I cannot write things half in terms of r and half in terms of x So what I'll do is I'll finally replace my r cos theta as x, you know r cos theta is x. So this can be replaced with x And since r sine theta is y this can be replaced with y Correct Right Now If I bring this term down over here Are you able to recognize this? Can you recognize this? Remember the locus definitions of conic sections Remember the locus definition of conic section. Are you able to relate it to this? Okay now now if I just write it like this find the locus of a point which moves in such a way That its distance from 2 comma 1 Is equal to its distance from x minus y equal to 0 now. Do you recognize it? Distance from 2 comma 1 is equal to distance from x comma x minus y equal to 0 Do you recognize that this is going to be this particular curve? So when the distance of a point is same as the distance from a line 2 comma 1 and this is x minus y equal to 0 So when distance of any point h comma k is same from this line and this point Then what is the locus of h comma k? What's the locus of h comma k guys? I'm waiting for your response It is parabola loud and clear the message is delivered Sai what are you doing man? Three times you change your answer but still could not get the right answer Guys you need to take care of your accuracy Don't look at what others are answering you stick to your concept. Let it be right or let it be wrong, right? It's such a beautiful question which actually helps you to revisit your conic section as well Okay, moving on to the next question That's question number 13 for the day more than one options can be correct in this question So please read this question very very carefully Am I audible? Can you see the screen now? Yeah, now it's visible to all guys any idea how to solve this problem So there's absolutely no problem with the other curve, right? Let's say I call this curve as C1 and I call this curve as C2 Right So C1 is plain and simple C1 is basically a circle This is nothing but a circle Whose center is that? 3 comma Minus 1 right So 3 comma minus 1 would be somewhat like this and its radius is also 3. So basically it's going to touch the It's going to touch the y-axis like this Right. So this is your C2 curve No problem regarding C2 Now what is C1 curve? So this is like saying argument of 3z minus 6 minus 3i minus argument of 2z minus 8 minus 6i is equal to pi by 4 So for the time being you can treat this as Treat this argument to be Alpha treat this argument to be beta. So you can say alpha minus beta is given to you as pi by 4 So you can do one thing also, uh, you can always pull out a 3 out from here That means you can write it as argument of 3 plus argument of z minus 2 minus i Minus argument of 2 minus argument of Let me write it down minus argument of z minus 4 minus 3i is equal to pi by 4 Is equal to is equal to pi by 4 So guys argument of 2 and argument of 3 are both zero So you can recombine it and write it as argument z minus 2 plus i By z minus 4 plus 3i is equal to pi by 4 Again this fellow reminds me of kony All right, it reminds me of the kony rotation formula So z is moving in such a way that the angle that it subtends at 2 plus i which is actually a point of 2 comma 1 If i'm not mistaken 2 comma 1 will be lying somewhere over here By the way, this is 1 so this is 2 so this will be even bigger circle Okay, let me just erase this circle and draw a bigger circle in place of this Okay, assume this to be circle so 2 comma 1 is somewhere over here. Okay And uh 4 comma 3 would be somewhere over here So this is 4 comma 3 point. This is 2 comma 1 point. Okay And there's a point z which is free to move in such a way that Again draw the kony for this so 2 plus i z 4 plus 3i And it should be like this No, this is not the right method because here you are getting in a clockwise sense Okay, but here the angle is positive. So it is this is not the right diagram Okay So try the other way around So you should have a anti clockwise. This is 2 comma i or 2 comma 1 and this is 4 comma 3 Then this is fine. Then this is pi by 4 Okay Yes or no Right, so how should it be positioned? It should be positioned in such a way that This angle here This angle here in an anti clockwise sense should be pi by 4 So it is freely moving on a circle So it is moving on a circle Like this I should have drawn it. Yeah, is that clear Now this circle This is a part of a circle This is a part of a circle Can somebody tell me the center of this circle? And can somebody tell me the radius of this circle? So think as if you have A circle like this and you have two points Sorry You have two points 2 comma 1 and 4 comma 3 Okay This is not there And you're trying to find out the center of this circle and radius of this circle Now this angle here This angle here is 45 degrees Right, so this angle here would be 90 degrees This would be 90 degrees Correct Now If I just bisect this line Okay Now this point will be 3 comma 2 This point will be 3 comma 2 Yes or no So this distance will be how much? This distance will be root 2 Right This distance will be also be root 2 So this radius will going to be 2 centi... 2 units So radius is 2 units Now what about the center? For finding the center We need to locate a point which is at a distance of root 2 from this point and lying on this line So for that you have to recall your distance form of a Distance form of a line Right Now if I ask you what is the slope of this dotted line? What is the slope of this line? Slope of this line would be negative reciprocal of slope of AB Correct So slope of AB let's find out first y2 minus y1 y2 minus y1 by x2 minus x1 which is going to be 1 So this line would have a negative slope of minus 1 so m would be minus 1 Right So try to recall the distance form of a line distance form of a line says x minus x1 By cos theta equal to y minus y1 by sine theta Is equal to r Right So this r is going to be root 2 And I'm going to figure out a point at a distance of root 2 from that point So this is going to be root 2. So what would be x? x would be This value will be minus root 2 because x should increase as you go towards the right. Okay So x would be Minus 1 by root 2 into minus root 2 that is going to be 1 Okay, x minus 3 is going to be 1 That means x is going to be 4 So center x coordinate will be 4 And y minus 2 is going to be y minus 2 is going to be 1 by root 2 into minus root 2 Which is minus 1 So that means y is going to be 1 So center will be 4 comma 1 that means your second circle would have an equation of x minus 4 whole square Plus y minus 1 whole square equal to 4 So we have both the circles with us now 1 is x minus 3 whole square plus y plus 1 whole square is equal to 9 And the other one is this so let us solve these two simultaneously Let us solve this two simultaneously To see where do they intersect or do we have any point of intersection for them? Right, so let's open this up first. So first equation will be x square plus y square minus 8x minus 2y 16 plus 1 17 17 minus 4 is going to be 13 equal to 0 And the other curve is going to be x square Plus y square minus 6x plus 2y Plus 1 equal to 0 Okay How do we solve this simultaneously? Subtract both of them. So subtract these two You will get 2x plus 4y minus 12 equal to 0. That means x is equal to You can write it as a 6 minus 2y Okay, and put this value in any one of the functions in any one of the curves Okay, let me put it in the second one. This one this is looks easier So we'll have 6 minus 2 y square plus y square minus 6 into 6 minus 2y plus 2y plus 1 equal to 0 Or let us simplify this it gives you 36 plus 4 y square minus 24 y plus y square minus 36 Plus 12 y plus 2 y plus 1 equal to 0 So 36 36 goes off Uh, you have 5 y square So 5 y square You have minus 12 y sorry minus 10 y Plus 1 equal to 0 Okay So when you use Sridharacharya formula you get 10 plus minus under root b square minus 4 ac minus 20 by 20 by 10 by 10 Okay Yeah, so it's a 1 plus minus root 80 root 80 is 4 root 5 So it is 1 plus minus 2 by root 5 Now clearly it cannot be uh It cannot be 1 plus 2 root 5 Because 1 plus 2 root 5 will give you The y coordinate greater than 1 That means it is coming from this part of the Curve Correct? It is coming from this part of the curve, which is not possible So we have only this part of the curve So one of the signs will be rejected That means I cannot have I can only have 1 minus 2 root 5 as our right answer Okay, so for this find the value of x x will be 6 minus 2 y 6 minus 2 y that will give you 4 plus 4 root 5 So we have one point of intersection Which is 4 plus 4 by root 5 Plus i 1 minus 2 root 5 so option number a is definitely correct Isn't it so option number a will be definitely correct Now let's see what does question number 2 say I'm sorry option number 2 say option number 2 says Complex number z in the set q for which this equation takes maximum value is minus 1 minus i Now guys again try to identify this What does this signify again in terms of locus look at it in terms of locus If you look at it in terms of locus It basically signifies the distance of difference of the distance from 2 comma 1 and 4 comma 3 It's saying the distance has to be maximum When can this difference of the distance be maximum? Any idea where should z lie says that the difference of the distance Should be maximum can I say it can lie anywhere On this line itself Let's say I extend this Then this distance will always be maximum and that will be equal to the distance between these two only Is that correct? So for distance to be maximum These three points should be collinear Yes or no That means x plus i y if I write z as x plus i y Then basically x plus i y should be the same should satisfy the equation of this line Right, which is y minus 1 is equal to slope times x minus 1 So y is equal to Sorry y minus 2 is y minus 1 is equal to x minus 2 So y becomes So y becomes x minus 1 Now I have to see that this complex number z That I should get Should be from the set q That means it is simultaneously satisfied the Equation of the second curve Isn't it? so this should also satisfy The equation of the second curve, which is that of a circle x minus 3 square plus y plus 1 square equal to 9 so we have to solve these two simultaneously Okay, so put y as x minus 1 over here. So it becomes x minus 3 square y plus 1 will be x square equal to 9 Open the brackets Open the brackets you get x square minus 6x plus 9 Equal to 9 so basically x becomes 3 And x becomes 0 If x becomes 0 y will automatically be minus 1 that means 0 minus 1 i is going to be the point which is going to be minus i itself So yes option number b is also correct Okay, guys, so we'll not spend too much time on this. Similarly. Can you verify c and about c? Okay, verify about c Okay, if I ask you what will be the minimum What are the minimum distance where should it be lying? Where should the complex number lie so that it has minimum distance from These two points two plus two comma one and four comma three Just a question You don't have any space to write Is my question clear? I'm trying to ask you when will be When will a point z have a minimum distance from these two points? So two comma one four comma three. Where should it lie that it has a minimum distance? You will say obviously it should lie on the bisector of bisector of these two lines Isn't it? Yes or no? Yes or no guys, is it clear? So you find the bisector equation and again simultaneously solve it with q to check whether this point satisfies it or not and let me know the Result for this whether c is the right answer or not. So I'm leaving it open for you guys That's was a huge question actually Okay, so we'll quickly attempt this All right. So in the interest of time, I'll be solving this for you In case you are already trying it avoid looking at the solution Now since it is mod z square, I can write it as z one plus alpha to the power r z two times conjugate of this So when you open this term when you open this term Let's see. What do we get? z one plus alpha to the power r z two times z one conjugate alpha conjugate to the power r z two conjugate So we'll give you mod z one square correct and it will give you z one z two conjugate alpha conjugate to the power r plus z one conjugate z two alpha to the power r plus mod alphas to the power two r mod z two square Now you're applying a summation on this term From r equal to zero to n minus one. Okay Now guys, uh, if you look at this term Mod z one square is independent of r. So it will be just summed n number of times So I can say n times mod z one square will come up correct, right and here if you see Mod of alpha is going to be one because alpha is a complex cube root of unity So this also is going to become one So it simply leaves you with mod z two square. So the summation of that will be n times mod z two square Now what about these terms? So this term is z one z two conjugate summation of alpha conjugate to the power r from r equal to zero to n minus one And the other term is z one conjugate z two summation alpha to the power r from r equal to zero to n minus one Now we all know that summation alpha to the power r from r equal to zero to n minus one is going to be zero that is the sum of the complex or the sum of the roots of Some of the n nth roots of Unity is zero is that fine So can I say even the sum of the conjugates will also be zero So can I say some of the conjugates will also be zero? It's very simple because if summation cos two r n Sorry two r pi by n plus i sine Two r pi by n is zero Then so would be the summation of cos two r pi by n minus i sine two r pi by n would be zero So even for this it will be zero for the same reason So these two terms Will vanish off leaving me only with this as the answer Which is your option number? Which is your option number? c Is that fine guys? So moving on to the last problem. So this I'm giving you one minute to think of it. It's a very super simple problem Again more than one option may be correct in this case So these series seems to be very familiar to me, right? So here something is fine, but here you see one omega is less So x cube omega square x to the power five omega to the power four Okay, so what I'll do is I'll just try doing a plus omega b I'll multiply this with omega And add it to a let's see what happens So I get one I get minus omega x I'll get plus omega square x square by two factorial I get minus omega cube x cube by three factorial plus omega four x four by four factorial and so on till infinity This clearly indicates it is actually the expansion of It clearly indicates that it is the expansion of e to the power minus omega x Right Yes or no, okay, and omega is the cube root of unity. Okay now We can also say that a minus omega b will give you e to the power omega x So adding these two I'll get two a is equal to e to the power omega x plus e to the power minus omega x That means a is going to be e to the power omega x plus e to the power minus omega x by two So option b is definitely correct Option b is Definitely going to be correct Now is there is there any other option which is correct if you multiply them If you multiply these two that means a plus omega b which is equal to e to the power minus omega x And a minus omega b which is e to the power omega x you multiply them What do I get a square? Minus omega square b square is equal to one Oh, so option a also becomes correct Right. So a and b would be correct in this case Okay All right guys. So with this we reached to the end of this session Uh, we had a good amount of problems all today 15 questions But more importantly these questions told you where you are weak at Okay, so locus base questions and a throat of unity base questions Rotation formula base questions you have to practice a lot Right So thank you very much for coming Live on youtube over and out from symptom academy. Have a good night. Bye. Bye