 So, application of complex numbers continued in geometry, sorry, application of complex numbers in geometry continued. Last class we had talked about equation of a complex line in the argon pane. We also talked about the concept of complex slopes, right? We talked about the complex slopes relation when the two lines are parallel and when the two lines are perpendicular. We also talked about the distance of a point from a complex line. Okay, so now we have to speak a bit about circles. In fact, I wanted to speak about ellipse, hyperbola, parabola as well. But most of you are aware that you have not seen the conic section videos yet. So, without that it will be very difficult for you to understand those concepts. So I will wait for the right time to come when you see those videos or probably when we do it in the class itself. Then I'll come back to those concepts. They are all linked to the same basic core principles. So, let us understand the concept of equation of circle in the argon pane, in the argon pane or in the Gaussian plane. So, what is a circle? As a locus definition, what do we know about a circle? So, circle is basically the locus of a point which moves in such a way. In a plane such that its distance from a fixed point, let's say there is a complex number z which is free to move and it is moving in such a way that its distance from this fixed point which is z0 is always fixed. That means this distance is a constant. Let me call this as small r. r is a purely real number because it represents the distance. Whereas z and z0, they can be complex numbers. So, when you look at such a scenario, what is the first thing that comes in your mind when you want to relate z, z0 and r? What comes in your mind when you want to relate z, z0 and r? Distance formula. Absolutely. Modulus formula that is synonymous to the distance formula. So, we'll say the first thing that comes to my mind is z minus z0 modulus is going to be r. And there you go. This is the equation of the circle. As simple as this. So, this is the equation of a circle. Equation of a circle whose center is at z0 and radius is r. If I modify this a bit, let's say I say less than r, then it will be interior of a circle. So, it would be a part which is within the circle. And if you say z minus z0 is greater than r, it will be outside the circle or exterior to the circle. So, many times you will see these inequalities while solving complicated problems. That time, please remember less than means within the circle, not on the boundary. For it to be within the circle or on the boundary, it should be less than equal to. Got the point? So, in the same way, on the circle and outside, it will be greater than equal to. Okay. Anyways, we'll talk more about this guy, which is the equation of a circle. Now, as you know, these J guys will not be happy when you just write or they will not be happy to see you satisfied with this equation. So, what they will do is they will complicate this. So, this can be complicated. So, another version of mod z minus z0 equal to r. So, what they will do is they will write it like this. So, first they'll square both the sides. Okay. We all know z minus z0. For that matter, any complex number's modulus squared is the product of that complex number with its conjugate. Correct? Now, let us open the brackets over here. This is nothing but z minus z0. z conjugate minus z0 conjugate equal to r squared. So, this will give you z, z conjugate. Okay. Minus z0 conjugate z minus z0 z conjugate plus z0 z0 conjugate. And this r squared, I'll bring it to the left side. Okay. Now, this is mod z squared. This I will write it in a slightly different way. Not very different. Okay. This also mod z0 squared minus r squared. Okay. Now, if you look at this structure, this structure is that of this. Okay. Where I have called this complex number, as you can see over here, I have called this complex number as A. So, this automatically becomes A conjugate. And this whole thing I have called it as B. Now, here you can see A is a complex number while B will be a purely real number. Because mod z0 squared minus r squared, both will be real in character. So, real minus real will give you a real number. So, this is the general form, general form of the equation of a circle in argon paint. Okay. So, of course, you would be very lucky if you see it in this form. I mean, the form which I wrote here on the top. But don't be surprised if somebody gives you the equation of a circle like this also. Okay. It is like saying somebody can give you an equation. You all have done a circle equation to a certain extent in school also. So, if somebody gives you something like this, x minus 1, the whole square, y minus 2, the whole square equal to 3 square. You will be more than happy to see this form because you can directly know its center, 1, 2. You directly know its radius, right? Which is 3. But if somebody gives you like this, then it's like not to be surprised. They have basically expanded this and given it to you. Okay. So, this is 1, 5, minus 4, I believe. Okay. So, this and this are basically the same thing, just written in different forms. Okay. Any question coming up? Good. Okay. So now, many times what will happen? They will give you the equation of a circle in this form and they will ask you the center and the radius for such a circle. So, how will you find center and radius? Very simple. When this form will be given to you, you would know your a. You would know your a conjugate. You would know your b. Okay. So, what you have to do is, just remember the simple comparison. Minus z0 is compared to a. So, z0 which happens to be your center is minus a. Okay. So, please make a note of this. This will give you the center of the circle. Okay. And similarly, mod z0 square minus r square is equal to b. So, your r square is mod z0 square minus b. Okay. So, r will be under root of this, which we can write it as mod a square because z0 is minus a, no? So, z0 mod is same as mod of a. Okay. Minus b under root. So, this is the formula for the radius. Okay. So, once you have been given the equation of a circle in the general form, you can easily figure out the center and the radius by using this simple formula. Copy. Some people are joining late. Hmm. Class was from 345. Yeah. So, those who joined in late, please copy this. Please take a snapshot if you want to because I'm going on to the next page. We'll do one small question on this. Very, very small question. One non-nomina question we'll take on this. Sir, I joined two minutes before 345. Very good, Akash. Proud of you. Who all joined late? Chukke se tell me so that you don't lose your attendance. Sir, I joined late, sir. Big mistake, sir. I didn't read the message. Yes. So, let's take... Okay, Mr. Dash. Karthik. Yeah, yeah, yeah. Karthik, you joined a little bit early. Yes, I know. Okay. Find the, find the kendra center and radius of a ayat circle. Sir, you are teaching in Hindi also, sir. 2zz conjugate plus 3 minus iz plus 3 plus iz conjugate minus 7 equal to 0. Guys, be very, very careful while assessing this. Okay. I'm asking for the center and the radius. Please give your answer like C, which complex number, R, what real number, okay, on your chat box. So, what is the center and what is the radius of this circle? Many people are absent. Well, they are traveling, going to your relatives' house. I can understand. It's the festive season. Yes. Anybody, anybody of heart who wants to disclose their answer? Please do so. Nobody. Is it so difficult? Come on. No, it is not. So, Pranav gave an answer. Pranav, sorry to say, but that is wrong. To your surprise, it is wrong. Pranav is wondering, how come just now he told this formula. Advik also, sorry to hear wrong. Guys, and ah, Aditya got it. She is wrong. Yes, there is a two sitting over here that is spoiling the business. Guys and girls, my dear students, the formula which I gave you that had a Z, Z conjugate over here, not two Z, Z conjugate, right? So, will that make my formula change? Yes, it will. So, if you don't want to change your formula, first thing is you have to rephrase it, rephrase it like this. Now, you can say this guy is your A, this guy is your A conjugate, and this guy is your B, okay? Please be very, very careful about that structure. You cannot afford to use the same formula if that structure has been distorted, which many a times will be distorted because the examiner knows you would have mugged up the formula without knowing, okay? Or without understanding it deeply, and hence they will try to trick you. Now, having converted it to this form, your A is going to be this, so negative A is going to be a center. So, minus 3 by 2 minus I by 2 would be your center. So, this will be your center of the circle, okay? Radius, as you already know, under root of mod A square minus B. So, it will be under root of mod A square will be 9 by 4, okay? Plus 1 by 4 minus of minus 7 by 2, that is plus 7 by 2, okay? Let's do one thing, let's multiply and divide with the 2 here and we can take 4 as the Lcm. So, it's 14 plus 115, 15 plus 9 is 24. Root 24, okay? Root of 24 by 4, which is root 6 units. This is the equation of the, sorry, this is the radius of the circle, my bad. Now it's correct, now it's correct. Yes, radius is a real quantity. You are giving me a complex number for radius, no? Radius is a purely real quantity. Radius signifies the distance of that freely moving point from Z naught. Radius, guys, again I'd like to re-trade. Center is minus A. Radius is under root of mod A square minus B. No, it will always be a conjugate. If it is not a conjugate, it is not the circle, first of all. It is not going to deviate from that structure. If it does, it doesn't represent the equation of a circle, but it is something else. Simple as that, okay? Good, now I will show you some different forms of the equation of a circle. So, other forms of the equation of a circle, okay? So, the first form, sorry, the second form that I would like to discuss with you is when somebody gives you the diametrically opposite points of a circle and asks you the equation of a circle. So, let's say this is Z1 and Z2 and they say what's the equation of the circle? Now, I'm sure all of you would be thinking, okay, what is the big deal in this? We can know the center. Center is Z1 plus Z2 by 2. Remember midpoint formula, section formula, etc. Whatever we have learnt in co-ordinate symmetry, that will be, again, applicable to complex numbers as well. So, you can simply say mod of Z minus the center would be Z1 plus Z2 by 2 equal to half the distance between Z1 and Z2, okay? While I definitely agree this is one of the, you know, answer that you would be thinking of, but this is slightly complicated to look at. There's another way to look at it through the concept of complex slopes. So, let's say I call this point A and I call this point B, okay? And let's say I call this point B, okay? Can I say if this has to represent the circle, okay? Then, then the slopes of these lines must be such that they must make a 90-degree, you know, in between them. So, let's say this has a slope of W1, this has a slope of W2. So, we know that W1 plus W2 should be 0 if it has to represent two perpendicular lines. And it has to be because then only P will lie on a circle. Correct? So, what is W1? W1 is nothing but Z minus Z1 by Z conjugate minus Z1 conjugate. That's what we had learned in our complex slopes. Complex slope is nothing but whatever two points are given to you, just take the difference of the two points and divide it by the difference of their conjugates, okay? Similarly, slope of BP. This is the slope of AP. So, the slope of BP will be what? Z minus Z2 and Z conjugate minus Z2 conjugate equal to 0. Okay? If we simplify this, it becomes Z minus Z1 times Z conjugate minus Z2 conjugate plus Z minus Z2 and Z conjugate minus Z1 conjugate equal to 0. So, this is another form of the equation of a circle and this is what we call as the diametrical form. Diametrical form of the equation of a circle. Okay? Please make a note of this. If you expand it, its structure will be the same as what we discussed in our center radius form. A little while ago. Is that okay? Any questions? Any questions? Any concerns? Right? Another way is, many people recommend, sir, why don't you use a coni here? Let's apply a coni at Z. Yes, we can definitely do coni at Z. Or why can't you apply your Pythagoras theorem? Pythagoras theorem, that also can be done. Okay, so let's look at another form. Another form is Z minus Z1 mod square Z minus Z2 mod square is Z1 minus Z2 mod square. Correct? So, basically I'm applying a Pythagoras theorem. Okay? This is another form. Okay, so let me name it. This is the second form. This is the third form. This is the fourth form. Correct? If you apply coni at Z, let's apply coni formula, coni rotation formula at Z. Formula at Z. Okay, assign some arbitrary direction to it. Let's say I take an anticlockwise direction. So, I can say Z2 minus Z by Z1 minus Z is equal to mod Z2 minus Z by mod Z1 minus Z e to the power i pi by 2. Now, what I will do is, I'll slightly make it more generic. How? I will introduce plus minus here. Why? Because then I will be able to cover points which are lying below the line also. See guys, if you take plus pi by 2, you will always get Z on the top arc. Remember we have done a question like this. Okay, but if you take plus minus pi by 2, you can cover up the below arc also. So, I should say that you can cover the entire circle. Are you getting my point? Right? Okay. Is it fine? Clear? No. This looks slightly complicated. So, there is an easier way to look at it. Many books will suggest this. Argument of Z minus Z2 by Z minus Z1 is equal to plus minus pi by 2. Okay? Or you can just simply reverse it also. Doesn't make a difference. Just to keep Z1 on top, Z2 below, plus minus pi by 2. This is another, yet another form of the equation of a circle, which is your fifth form. Okay? So, see so many forms I am showing you. Okay? All of them convey the same fact that you are finding the equation of a circle whose diametrically opposite points are Z1 and Z2. Right? Whether you come from the concept of complex slopes, whether you come from the concept of center radius form, whether you come from the concept of Pythagoras theorem, whether you can come with the concept of the ponies rotation formula, all of them basically end up giving you the same, I can say, circle, but in different, different forms. They are different forms of the same thing. Okay? Just like you learned so many forms for the equation of a straight line in 2D, right? 2 point form, slope point form, intercept form, normal form, distance form. They were all talking about the same line, but different, different ways. Isn't it? Okay? Now, I will generalize this last one a little bit more. Right? This part that I have discussed over here, I will generalize this a little bit more. If somebody says that there is an equation which looks like this, I'll point towards this so that you understand when you're reading the notes. If somebody gives you the equation of a point or equation of any curve like this, equal to some alpha. Okay? Alpha is some angle. Remember, this is nothing but it will show you an arc of a circle. Okay? It will give you an arc of a circle. How? Very simple. See, when you read this, when you read this, what comes in your mind? Basically, I get to think of or I get to imagine that there is a point z where if I write the Coney rotation formula applied to z as the pivot point. Okay? Then this angle, then this angle is always, this angle is always an alpha. Now, alpha can be positive, alpha can be negative. That depends upon which of the two arcs because if two points are there on a circle, it will divide the circle into two arcs, major and minor unless until they are the diameter. Okay? So in this case, let's say I'm assuming alpha to be positive as of now. It could be negative also. Okay? Then this would represent this arc, the one which you can see on your screen with the dotted, but there would be holes over here. There would be holes over here. So let me punch a hole. Right? Why holes over here? I've already explained this to you in the previous class because if you include your z as z1, this entire thing will become a zero, won't it? So you're trying to say argument of zero is alpha, which is wrong. Argument of zero is not defined. Correct? And if you're putting your z as z2, this whole thing becomes undefined. Right? That is why I have punched a hole at z1 and z2. So this would represent, this equation would represent this arc. Okay? If alpha is positive, it will show you the upper arc, alpha is negative, it will show you the below arc, but some arc of the circle. Right? So if this alpha becomes plus minus pi by 2, or let's say plus pi by 2, then z1 and z2 will actually become diameter ends. Correct? Are you getting my point? Let's say this alpha is lesser than pi by 2. Then z1 and z2 would become ends of a chord. Right? And the same chord will actually become a diameter if alpha is 90 degree or minus 90 degree. That is what I wrote over here. Let me remove my camera. That is what I wrote over here. If this alpha becomes plus minus 90, right? Then it will cover the whole circle. Okay? But still in this, z1 and z2 will not be covered. Right? So even if you're making a circle, make sure that you make a small change in the diagram. I have to punch holes over here. So there will be holes. So my z cannot become z1 and z2. Are you getting my point? This type of questions are very, very commonly asked. I think I should take a question on this form. Let's take one question. By the way, if you want to copy something, please do so. I'm going to the next board. Major, minor arc if alpha, yes, absolutely enough. If alpha is more than pi by 2, it will represent the major minor arc. And if it is less than pi by 2, it will represent the major arc. Absolutely well assessed. Good. Should we go to the next page? Should we go to the next page? Okay, Gayatri. Let's go. Finding the right question is also a great question. I hate this. I hate this searching for questions. Did I do this question with you? This one? Anybody remembers this question being done in the previous class? No? No, sir. Okay, fine. So let's do this question. But I know this question will be not too easy for you to answer because you have just not done the theory. So I'll make it slightly complicated. Okay. I want you to give me the center and the radius for the circle as well. Okay. I know all of you will go for circles. So even I'm not putting the pole because 100% response, I'm sure. Okay. 100% of you will say circle. Understood. It's the circle. But I want you to tell me what would be the center and the radius for such a circle? Okay. Assignment it was there. Okay. Find the center and the radius of the circle. Okay, Pranav. Good. Not saying right or wrong to it. Even I think so. You thought it is pi by 2. It is pi by 4. Anybody. Okay, Aditya. That seems to be correct. Okay. Let's discuss this. Let's discuss this. So first of all, what I'm going to do is I'm going to just give it a slightly different shape. Can I write it like this? 1-z by minus 1-z equal to pi by 4. Will it make any difference to the expression? I don't think so. Now somebody may ask me why did you do like this? What was wrong with this? See basically I wanted it to give it a form of the Koni rotation formula where rotation is happening at a pivot point z. Okay. Remember pivot point is always in the second position. Right? So minus z minus z is what I wanted to make. That's fine. Right? So let's say 1 and minus 1. 1 and minus 1 will be here. 1 and minus 1 will be here. Right? Now let me take a z. Very good. Let me take a z over here. Okay. Now what I'm trying to claim is if I connect, see all of you please watch out this move. Okay. If I connect this. Right? And if I try to apply Koni taking z as the pivot point, remember I have to make anti-clockwise plus pi by 4 means anti-clockwise. With the arrow at 1 and the tail at 0. But the moment I do this, it becomes a clockwise angle. Right? You just said this is clockwise. This is not anti-clockwise. So this angle should have been minus pi by 4. So either this question is wrong or your diagram is wrong. Right? This is what you will say. Obviously question cannot be wrong. My diagram is wrong. This is what I was trying to say. If this changes its sign, you will start getting different sides of the circle or different arcs of the circle. So this diagram is incorrect. And in your GE question or in your cognitive level exam question, they'll say upper part of the circle, lower part of the circle, like that they will mention in the answer. They will not just mention circle. Okay. They are too smart for that. Okay. They will try to test you whether you know which part of the circle, which arc of the circle you are trying to plot. So this is wrong. Right? So I will redraw this by putting my Z on top. So Z cannot be here. Z has to be somewhere over here. Okay. That's it. Now when I connect it, now when I connect it. Okay. Then arrow head should be on one. So this is my arrow head. And arrow tail should be on the other one. And this should be anti-clockwise pi by 4. Is it coming anti-clockwise? Yes. Very much. Okay. So this is the correct diagram. Okay. So Z will be moving on this part, my dear. Z will be moving on this part of the circle. Of course, there will be a hole over here. Okay. But my concern is to find the radius and the center. Now, center we all know would lie on the perpendicular bisector of this chord. So this will be like a chord. Right? So center will lie somewhere on the imaginary axis. Correct? So let's take an imaginary axis point. Zero comma Y. Okay. Now how do I get Y? Okay. If this is pi by 4 and if I connect my center with the end of the chord like this, then this angle should be 90 degrees. Correct? This angle should be 90 degrees. Correct? And not only that, this is 45, 45. So if this is one, this also has to be one. This also has to be one. So if I draw this structure over here, just all of you please pay attention. So see what was happening. This was 90 degree. So when I drop a perpendicular from here, which is basically your origin point. Okay. This should be of the same length as this length. Okay. And of course this should be also the same length because this is 45, 45, 45, 45. Okay. So if this is one, this is origin and this is minus one comma zero. So if this length is one, so will be this length, right? So y will be nothing but one here. So this y here, sorry for coming in between the lines. So this y here will be zero comma one. That means your center would be at i. Okay. So this answer would be i. Zero comma one is what? Zero comma one is i only, right? And what are the radius? Radius is going to be the hypotenuse length. So this is one, this is one. So this has to be root two, root two units. Okay. So a direct question may be asked to you that if there is a curve that satisfies or if there is a circle whose equation is argument z minus one by z plus one equal to five by four. Then which of the following option is correct about the radius and the center and this would be one of your options. So you should definitely mark it. Is it fine? Any, any cluster? Could you explain why the first diagram was wrong? Okay. Very good question. See, if you take minus one comma zero and one comma zero like this. Okay. And if you take a z below. Okay. Now read this expression very, very carefully. This expression very, very carefully. Read this very, very carefully. It says you are applying Coney at z. All right. Where the angle direction has been chosen in such a way that the arrowhead of the angle is hitting at one. So it is somewhat like this. This is the arrowhead hitting at one. That's why one minus z is coming on top and arrow tail is at the other line which is connecting minus one comma zero and z. So that is why it is there in the denominator and this should be an anticlockwise direction of anticlockwise direction of five by four. But is it anticlockwise? Is this direction anticlockwise? Right? It is not. So if it is not means this diagram is wrong. So is there should be up, not below. Are you getting my point? So these things are very small things which people ignore and that may cost you the entire problem. Right? Okay. So without much waste of time. Sir, we are not wasting time. Sir, we are studying. Okay. Without much ado, let's look into the last form of the equation of a circle which is the fifth form. And this is slightly surprising to many of you. Another important form. Another important form. Mod z minus z one. In fact, let me keep my mod small. Mod z minus z one by mod z minus z two is equal to K and K not equal to one. This also represents a circle. This also represents a circle. Okay. Now, many people ask me why not equal to one? What was the problem with one that you removed one? Right? So let's say if your case equal to one, that means you're trying to say mod z minus z one is equal to mod z minus z two. Means basically z is lying in such a way that its distance from two fixed points z one and z two are equal. So it will lie on the perpendicular bisector of z one and z two. Right? So z one z will lie on the perpendicular bisector. So it will give you a line, not a circle. Right? So these two lengths are equal. That's what you're trying to say, right? By stating this equation, you're trying to say this length is equal to this length. Correct? So z is moving in such a way that it is maintaining a fixed distance from or it is maintaining the same distance from z one and z two. So how should it be? It should be on a straight line. So that's why it is not a circle. Now here many people argue, sir, can't we say it's a circle of infinite radius? Yes, it can be said to be a limiting case. Right? So this is the circle of infinite radius. Got the point? But many a times from the J, but many a times from the J paper, they would remove this z equal to one case and they will say it is a circle for which of the following, then one of the options will say K should be real number other than zero, except zero, sorry, except one, except one. Okay. This is one you should be marking. Now, I know most of you would contest that even for one, I should get a circle of infinite radius. But why to take Punga? Let's not take Punga from J people. They might mark it wrong also. Okay. So let's not, you know, go against their understanding of the concept. Okay. Now, why does a circle when K is not equal to one? That is something which I will leave up to you. Okay. But I will leave you with a small question also, not only prove that it is a circle, but I will also ask you to figure out this simple thing. It will take a little bit of time. That's why I'm not doing it right now. This circle will basically be a circle whose diameter will be such points Z3 and Z4 where Z3 is a point which divides Z1 and Z2 in the ratio of K is to one internally and Z4 is such a point which divides Z1 and Z2 in the ratio of K is to one externally. Are you getting my point? Again, this is something which I would like you to prove for homework. Prove that when you have this circle, this circle is basically this circle whose diametrically opposite points are Z3 and Z4 where Z3 is the internal bisector of the joint of Z1 and Z2 in the ratio K is to one and Z4 is the external bisector of the same point Z1 and Z2 in the ratio K is to one. So as to say that Z3 and Z4 divide harmonically. They are harmonic conjugates. Pranav has already understood this well done Pranav but I would request everybody to figure this out and there is a special name given to such a circle. This circle is called the aplonious circle. So please figure this out. Very easy but it is lengthy. You have to write a lot of things. You have to simplify it and all the stuff which I'm not doing because this is not a very important concept but you should be aware of it. Ah Hariharan, didn't you do section formula DPP that I had sent when I was doing straight line? There was a word called harmonic conjugates. Anybody who remembers harmonic conjugate word in DPP? No, people are not doing DPP seriously that means. Do one thing, after today's class just pull out the very first DPP of straight line chapter where the topic name would have been the review of Cartesian coordinate system. There there was a problem. I think I don't remember the problem number. I don't mug up things where they talked about harmonic conjugates. So basically these two points let me call it as p and q. They are called harmonic conjugates when they divide the same set of points in the same ratio but one internally and another externally. Okay, got it Hariharan. Now you should just pay attention to that sheet that had a question. The word is harmonic conjugates p and q are harmonic conjugates of each other. Anyways, so with this we will wrap up few other important things. I think you already know about it. So let's ask few questions to you. Let's take a question. If there is a point Z which is free to move, so Z is a moving point. Z is a moving point and there are two fixed points Z1 and Z2. Z1 and Z2 are fixed points such that mod Z minus Z1 plus mod Z minus Z2. Is equal to mod Z1 minus Z2. Okay, then Z lies on option A, the line segment joining Z1 and Z2. Option B, line joining Z1 and Z2 away from Z1. Okay, or you can say towards the side of Z2. Line joining Z1 and Z2 away from Z2 or on a circle. On a circle with Z1 and Z2 as diametrically opposite points. I'll put the poll on for this. I would like to see your response. Poll is on. Four people have responded so far. This is so difficult. I thought everybody will give me the answer quickly. Okay, let's wrap this up in another one minute. Last 30 seconds. Okay, five, four, three, two, one, go. What, what, what, what, what guys? 10 of you have voted. What is this? Are there 11 of you? What are you doing? Vote, my sir. Vote. So much of pleading people. Two people are voting after so much of it. Everybody is in festive mood, sir. Some of you would be traveling also. Anyway, see the response. If I wear a person sitting on the hot seat of KBC, I'll get a heart attack saying this is fun. Maximum Janta has gone for B by the way, just by one vote, but there is a confused response. See guys, what are the meaning of modulus? What are the meaning of mod Z1 minus Z2 distance between Z1 and Z2, right? So what is this trying to say? It is trying to say that distance of Z from Z1 plus distance of Z from Z2 is same as distance of Z, distance of Z1 from Z2. A distance between Z1 and Z2. When can this happen? Can a point be here? Let's say, see, look at the first options. Line segment between Z1 and Z2. Let's say I take a Z over here. Can it be here? You'll say, okay, let's check. Is this distance plus this distance equal to this whole distance? Of course, yes. So first option is clearly correct. Right? But at the same time, I would like you to understand the second option, third option also. D obviously cannot be there. If it is square, square, square, then of course I can think of. But D is definitely not the right option. Now, if Z1 and Z2 is lying on this, let's say Z is lying here. I'll take another situation. Let's say if I extend this. So if Z is lying here, then what will happen? Then in that case, this distance minus this distance would have been this distance. Are you getting my point? So in such case, my equation or the locus equation would have become Z minus Z1 mod minus Z minus Z2 mod would have been Z1 minus Z2 mod. Okay, for this, this will be the situation. Are you getting my point? Okay, so there should have been a minus over here. Then only second option would have been correct. Got the point? In a similar way, look at the third option. Line joining Z1 and Z2 and away from Z2 means on the other side. So let's say this is my Z1, this is my Z2. And let's say Z was lying on this line extended over here. So let's say this is your Z. Then this distance minus this distance would have been this distance. So in that case, your equation would have become mod Z minus Z2 minus mod Z minus Z1 equal to mod Z1 minus Z2. So similar looking expressions, a small change of sign here and there can make a different locus altogether. So guys, be very, very careful when you're reading this or when you're geometrically trying to interpret the locus. This is not a thing to be mugged up because infinitely many situations can be created by using modulus and the argument combination. Okay, be very, very careful with such questions because they are the hotcakes. They are going to definitely come. Are you getting my point? Any questions here? Any questions here? Okay, now I would love to discuss other concepts also like the equation of a parabola, ellipse hyperbola, but most of you would not have seen those videos. So I would take that up in the right time comes. So with this, I'm wrapping up the complex number chapter and now I'm going to start with the new chapter.