 The next concept that we are going to talk about is the geometrical application of complex numbers. Geometrical application of complex numbers. Or you can say application of complex numbers in geometry. Many people say it is the application of geometry in complex numbers. Some people say it is the application of complex numbers in geometry. Both are actually the same. So let's try to discuss this concept. So few of the concepts that you already are aware of is the concept of distance formula. So how is the distance formula that you have learned in Cartesian coordinate system? Same as the concept of the modulus concept that you have studied in complex numbers. So distance formula, let's say I want to find out the distance between x1, y1 and x2, y2. So you all know that the distance is given by under root of x1 minus x2 square y1 minus y2 square. The same thing we can also use by, can achieve by the use of complex numbers by using the modulus operation. So modulus z1 minus z2. So if you do z1 minus z2, you end up getting x1 plus iy1 minus x2 plus iy2. So which is nothing but x1 minus x2 plus iy1 minus y2. And if somebody says, hey, what's the modulus of this complex number? You will say it is x1 minus x to the whole square. That is the real part square plus the imaginary part square, isn't it? So doesn't this match with this? So these two terms are analogous. So one is in geometry, other is in the use of complex numbers. The next thing that we're going to talk about is the section formula. In section formula of Cartesian coordinates, you know that if there is a join of two points, let's say x1, y1 and x2, y2. Let's say a, b are the two points and there's a point c which divides this in the ratio of m is to n. So let's say this is x3, y3. Then x3, y3 is basically given by m times x2 and times x1 by m plus n. This formula can also be applied even if it was a case of external division, correct? Now the same thing if let's say if you consider this to be z1 and this to be z2 and this complex number is z3. So you can directly say z3 is mz2 plus nz1 by m plus n. So whatever formula you have seen in your Cartesian coordinate system, they can be brought down also to the complex number. And in fact it is a very, you can say, efficient way of expressing it because as you can see, here you are using two coordinates. Here you are trying to deal it with a single expression. Of course, z1, z2 themselves are having your xy information into them. So it's very much same like what we have in the case of vectors as well. Now we are going to apply this also to learn equation of a line, equation of a circle, etc. So let's start with a line. So this is going to take the major amount of our time in today's discussion. Equation of a line, okay? Connecting two points, z1 and z2. So let's say I have a line which is connecting two complex numbers, z1 and z2. Okay. I want to get the equation of this line. How will I get this? So I want to get the equation of a complex line. So I would like to add the word complex over here. Okay. Now remember, equation of anything is basically trying to relate a generic point on that line with the other two known points. See, what is equation? When you talk about equation like 2x plus y equal to 3, what does it mean? It means any x comma y on that line will be linked to each other or related to each other by this relation. Isn't it? So equation of anything is just a relationship between an unknown point on that particular curve. It may be a curve, right? Need not be always aligned. So unknown point with a known information. So any kind of a relation which connects an unknown point with the known set of information that would be called an equation. And there can be several types of equation that can be formed. That's the main reason why when we are studying an equation of a line in 2D, there were several types of equations we learned. Parametric form, slope intercept form, normal form, two point form, slope point form. So depending upon the information that I have, we can make a connection between the unknown point and the known information. In the same way here, I would like to know, can you connect z1, z and z2? If yes, how? Something that you have learned little while ago which will help you to do that. What is slope here? Have I ever talked about slope in complex numbers? Can you be more specific, Madhav? What theta are you talking about? Yes, aditya. Basically we are going to talk about coni. We are going to talk about coni applied at one of the points. So let us see, let us apply coni at this. Let's say take this as a pivot point. And let's say I take this angle to be pi. Now I have purposely chosen an anticlockwise direction. So let us write down the coni. What will you write for this? You will write 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. Now if z1 and z2 were on the same side, let's say I take another situation. So you say sir, you have chosen it in such a way that it is coming out to be pi. What if your z was here, z1 was here and z2 was here? Then in that case the same equation will become this. Now don't worry, I am not giving this as an equation to you. I will simplify it in a different way. So I will get this in that case. Now both of them basically suggest you one thing. They both have one thing in common. They both have one thing in common and that one thing in common is they are actually suggesting you that z2 minus z by z1 minus z is a purely real number. Am I right? So if it is like this, it will be a negative number. If it is like this, it will be a positive real number. But they are real numbers. Isn't it? Now if you agree with this that z2 minus z by z1 minus z is a purely real number. Then for any real number, we have also known that. So let's say if z is a real number, capital Z is a real number, then z is equal to its conjugate, correct? If z is purely real, this also you have learnt in conjugate, correct? Now this will form the basis for getting the equation over here. So I can say z2 minus z by z1 minus z is equal to its conjugate, correct? So in conjugate, we know the property that, right? Now let's cross multiply. Let's cross multiply. So z2 minus z, z1 conjugate minus z conjugate is equal to z1 minus z, z2 conjugate minus z conjugate, okay? So if you multiply it, you get z1 conjugate z2 minus z, z1 conjugate. Then you will have minus z conjugate z2 plus zz conjugate equal to z1 z2 conjugate minus z conjugate z1 minus zz conjugate plus zz conjugate, okay? Cancel this out because they are same terms, okay? Now let us try to bring like terms together, okay? So let me bring everything to the left side. So if I bring everything to the left side, let's take z common. So z will give me z1 conjugate. In fact, I should write it like this. Let me write z conjugate common. So z conjugate, yeah, so z conjugate is here. So on this, if you bring this to the other side will be z1 minus z2, okay? And then take z, okay? If you take z common, you will have z2 conjugate minus z1 conjugate. And if I take this term, I will have z1 conjugate z2 minus z1 z2 conjugate equal to 0. Correct? Any questions here? Is this fine? Any questions, any concerns? Okay, now all of you know determinants, right? All of you know determinants? Anybody who doesn't know determinant? Okay, if you know determinants, you can actually write this as zz conjugate 1, z1 z1 conjugate 1, z2 z2 conjugate 1 equal to 0. You can easily verify this, right? So let me expand this for you. If you want, let us verify this. So if you expand this, you get zz1 conjugate minus z2 conjugate minus zz1 minus z2, okay? Plus z1 z2 conjugate minus z1 conjugate z2, okay? Multiply it throughout with a minus sign. When you multiply it throughout with a minus sign, you will get z conjugate z1 minus z2. Plus z, flip the position of these two, z2 conjugate z1. And again flip the position of these two, you will get z1 conjugate z2 minus z1 z2 conjugate, okay? Isn't this the same as this? Now why I have given you this determinant form because it is easy to remember. Nobody including me can remember this ugly formula of the equation of a line in complex domain, okay? Even I don't remember it, trust me. But I remember this, right? This is easy to remember. This is very difficult to remember, okay? So first thing is note this down. Then I will tell you something more about it. First note this down. One minute needed, given. One minute given. Any questions? No, in this case the order of z1 z2 will not matter. No, because I have taken an all-encompassing situation. So whether z1 z2 are on the same side, see z is not known to you. z is any gender complex somewhere on that particular line, right? So you don't know where is your z, right? So if your z is somewhere in between, you will form this equation. If your z is somewhere here, you will form this equation. If it is somewhere over there, you may have another equation coming out. All of them will basically tell you one single thing. That z2 minus z, z1 minus z is a purely real number, okay? That is what is important. Getting my point. If you have copied this, can we move on? Done. Okay, good mother. Thank you. All right, so I'll repeat this equation once again on the next page. Better copy pasted because let's copy this. Now I'll show you something very interesting, which is important for you to know. See, if you multiply with the i throughout, let's multiply with an i throughout. So let's say I multiply with an i throughout. Now why I'm doing it, I'll tell you the reason. When I'm multiplying i here, I'll just keep this as minus i z1 conjugate minus z2 conjugate. What I did basically was I flipped the position here so that I can bring out a minus sign. Now all of you please listen to this. Let's call this number as a complex number a. So let a be i z1 minus z2. Do you agree that if you're calling this as a, then this will be called a conjugate? Yes or no? Anybody who is doubtful about it, you should not be actually because if you take a conjugate of this, it will be i conjugate z1 minus z2 conjugate. I conjugate is minus i z1 minus z2 whole conjugate is z1 conjugate minus z2 conjugate. And one more thing I would like to highlight over here. If you see this number, I'll write it down over here. If you see this number z1 conjugate z2 minus z1 z2 conjugate. Basically you are writing z1 conjugate z2 minus z1 conjugate z2 whole conjugate. Aren't you? That means you are subtracting a complex number from its conjugate. When you do that, you actually get i times 2 imaginary part of z1 conjugate z2. Am I right? Correct? Yes or no? Now if you multiply this with an i, if you multiply this with an i, basically you are getting i square. i square is minus 1. And such a number is always purely real. Remember imaginary part doesn't mean it's an imaginary number. Imaginary part is still a real number only. So when you say x plus i, y, x and y are both real numbers only. Correct? So can I say this is actually a purely real number. So normally we say the equation of a line looks like this. a z conjugate a conjugate z plus b equal to 0. So this is the general form of the equation of a line. Equation of a complex line. Where a is a complex number or a may be a complex number. But b is a purely real number. But b is a purely real. Okay, is it clear? Any questions? Any concerns? See mother, here you are basically writing. First of all are you convinced with that this expression is same as this expression? That means you're doing a complex number minus this conjugate. A complex number minus this conjugate will be nothing but i times 2 imaginary part of that complex number. And when you're multiplying i throughout you are basically left with minus 2 times the imaginary part of a complex number. Which is actually a purely real number. That is what I'm calling it as a b. Now this a and b terms that I'm using is just to show you the general structure. Just like we say A x plus b y plus c. In equation of a line in 2D what did we used to say? This is a general form of the equation of a line. In the complex number this is the general form of the equation of a complex line. In order to show this that's why I did all these tantrums. Got the point? Is that clear? Isn't this purely real number? Isn't minus 2 imaginary this is a purely real number? Isn't this a purely real number? No, mother, mother. Pranav it is clear, I know. Mother. Or just because you saw imaginary here it becomes imaginary number. See, this takes you back to the first class of complex numbers. When you say x plus i, y, x and y both are actually real numbers only. So y is basically called imaginary part of z. But it is actually a real number. So if this is a real number can I say the structure of a line will look like this? That is what I'm trying to say. B has got no special meaning. B has just got, B is just like, you know, you have to know that this is the meaning. What do you know? Okay. Now one more thing that you should all be knowing here is that there is something called real slope and a complex slope of a line. So a line or a complex line has got two types of slopes. It has got two types of slope. One is called the real slope. Another is called a complex slope. So real slope is nothing but let us say these two points. Okay. I have to make a diagram once again. So let's say the two points that you were connected, let's say z1 and z2. If this was x1 comma y1 and this was x2 comma y2. Then the real slope is the one which you have already studied in Cartesian coordinate system, which is x2 minus x1 by sorry, y2 minus y1 by x2 minus x1. Okay. And there's one more thing complex slope which you are learning today that is given by minus a by a conjugate minus a by a conjugate. Basically very much like the way you find the slope of the general form minus a by b. So here you can say that x role is being played by z conjugate. Y role is being played by z. So minus a by a conjugate that will give you a complex slope. Okay. So there's a difference between the real slope and the complex slope of a line. Real slope is the same way as you find the slope of a line in Cartesian coordinate system. But the complex slope basically comes from the fact that it is the coefficient of basically it is made up of complex numbers. So this is a complex number in itself. So this is made up of minus a which itself is a complex number by its conjugate. Okay. Now many people ask me, sir, what is the significance of complex slope? Real slope we understand. It is the rise to run. Okay. Now a complex slope basically. Oh yeah. This is what Aditya is also asking. What is the geometrically difference between them? See complex slope is a phenomena which is not very well explained by many books. Even when I was studying, I was like, you know, very inquisitive about what is a complex slope. So I figured out certain discussions between a lot of mathematicians on math, stack exchange and all those things. So those people, they try to explain it in the form of a simple example. Let's say when you see a flight taking off. Okay. Let's say you are here. So you are watching the flight taking off. Okay. So you see that there's always a rise to the run, right? So this ratio is basically what you see is your real slope. This is your real slope, right? But actually when the flight takes off, it actually glides in this direction also, right? I mean, just to show you, let's say if I look at top view. So let's say this is your side view. Let's say if I show you a top view. So when the flight is taking off, it actually takes this route. Okay. So when you are watching it from here, let's say you are looking from this angle, right? You are basically seeing how much is the rise to the run, right? But actually there's a glide also in the, there's a glide also in the flight. Okay. So when the flight takes off, it doesn't take off like this. It takes off sliding also. It is moving forward, upward and left or rightward, whatever. So both IJK motions are there, isn't it? So when you try to see what is the glide with respect to the run, you are basically commenting on its complex slope, which is not visible to a realize. Are you getting my point? So glide to the run will be your complex slope and rise to the run will be your real slope. So what you see from here is the real slope because you can see it. It's a real, right? What you cannot see is becoming complex there. So what is hidden from you? That is the glide of the flight is hidden from you, right? So when I'm standing on the ground and watching the flight take off, my eyes cannot make out that it is moving away also from me, right? I can always see rise to that ratio. Getting the point. So the glide to the run is basically your complex slope. This is what I can explain you in a simple way, right? However, I will also send you some study material about the complex slope. If you find some time, you can read about it. So there has been some article written by some of the mathematician. Okay, anyways. Now, yeah. Now, in terms of a real slope is given by negative real part of a by imaginary part of it. Okay. And complex slope is given by this. So please make a note of this. So real slope is this while complex slope is this. Okay. So as you can see, it is completely real because real of a is real imaginary of a is also imaginary. So please make a note of this. Okay. Now why? So basically, I'm trying to say that this term is equal to this term. Can you prove it first of all? I can see some questions coming up. Complex thing cannot be plotted at all. You're trying to make it real. You're trying to give it a shape of a real thing so that we can study it. Right? For example, if I really talk about a complex thing to give you an example, what would be the length of a tangent or what are the length of a common tangent to a circle which is within the other circle? Can you draw a common tangent? First of all, you'll say no. But the answer is yes. There is a non real tangent between them which is common to each other. You cannot draw it. Hence you cannot figure it out. So complex soap of the line cannot be seen because we are trying to create a real scenario out of a complex scenario and trying to study it. As we will not be able to study it. Are you getting my point? For example, complex power which is basically stored within conduct, inductors and capacitance cannot be measured by a wattmeter. But that doesn't mean these electrical instruments do not store power in them. They store it in the form of something which cannot be seen by a wattmeter. So your eyes is like wattmeter. You can only measure real power. You cannot measure imaginary powers. But they are still there. They are stored in the conduct capacitors and inductors. Right? So real slope can be shown. Real slope you can say yes. I can see it. Complex slope cannot be shown. Right Aditya? The moment you are able to plot something it loses its complex. It loses its imaginary nature. It's a weird thing. That's why I told you will find these concepts slightly weird. By the way, one quick disclaimer I would like to give. These concepts will not be tested in mains. This concept will be tested in advance. So if you are a hardcore main aspirant, then you did not pay a lot of emphasis on this. But if you are a J advance aspirant, then of course you have to study this. Because they can give this as a comprehension-based question. Okay, ideas. Without much space of time there are a lot of things you have to cover. So how do I show that this term is same as this term? Very simple. You already know our A. What was our A? What was our A? I z1 minus z2. Right? So I z1 minus z2. This was our A. Correct? If you write this in the form of x and I y, this is what you have written. Or this is what you will write. So this is x1 minus x2 plus I y1 minus y2. So this is minus y1 minus y2 plus I x1 minus x2. Okay? So if somebody says, hey, what's the real negative real part of A? By imaginary part of A, what will your answer be? You'll say minus of minus y1 minus y2 by x1 minus x2. That is nothing but y1 minus y2 by x1 minus x2, which is nothing but y2 minus y1 by x2 minus x2. Okay? So in terms of A, in terms of A, you can basically say this is going to be the, going to give you the real slope. Okay? Is it fine? In terms of z1 z2, you can say it is nothing but, it is the real part of, negative real part of I z1 minus z2 by imaginary part of I z1 minus z2. That's what we can say. Okay? Is this fine? Now complex slope, there's another formula for complex slope, which is minus, sorry, which is z1 minus z2 by z1 conjugate minus z2 conjugate. So in terms of the two points that you know on the line, you can also write the complex slope like this, z1 minus z2 by z1 conjugate minus z2 conjugate. Now how, that is also very easy. Can I write A as I z1 minus z2? Can I write its conjugate as minus I z1 conjugate minus z2 conjugate? Correct? So this is your A, this minus sign is your, this external minus sign. This is your A conjugate. So automatically minus I minus I gets cancelled and you are left with this expression. Okay? So please make a note of this, especially the complex slope. Is it fine? Okay. One quick question I would like to ask you. Very quick question. Find, find the equation of the line, equation of a complex line joining, joining 1 plus 2 I and 3 minus 4 I. Let's do this quickly. Done? Okay. Now I'm sure most of you would not remember that complicated equation of a line that we had derived, but of course you know this. That's why, that's the reason why we discussed this. So z1 is this, z1 conjugate is this. Now z2 is this, z2 conjugate is this, 1 equal to 0. Just expand this. This will give you your answer. Okay? Now when you're expanding it, let's see what will happen. z, this minus this. So let me just write it directly. You'll get minus 2, minus 6 I. Okay? Plus z conjugate, this minus this. You'll get 2 minus 6 I. Okay? And when you do this, minus this, you'll have 1 plus 2 I, 3 plus 4 I, minus 1 minus 2 I, 3 minus 4 I equal to 0. Okay? Let's try to simplify this. Okay, so this will give you 3, 3 plus 8 I squared. 3 plus 8 I squared is minus 3, which is minus 5. Okay? 3 plus 8 I squared is minus 5. Correct? And you have 6 I and 4 I, which is 10 I. Okay? Similarly, here I'll get minus 3 plus 8 I, which is minus 5 again, and you'll end up getting minus 10 I. Okay? So now this will become z, minus 2 minus 6 I, plus z conjugate. Okay? And you'll get a plus 20 I. Okay? Now, many people start making a lot of hue and cry over here. They say, sir, you said this will be a purely real number, but it is a purely imaginary number. Wait. Before that, are these two the conjugates of each other? Okay? I think in one of the batches, I think it was in Raja Ji Nagar or something. When I give this, somebody started jumping. Sir, you said this is purely real number. It should come out to be like, a conjugate z plus a z conjugate plus b equal to zero, where this is purely real. But this is purely imaginary. But first of all, are the coefficients of z and z conjugate themselves conjugates of each other? No, right? They're not conjugates of each other, right? So what I'm going to do is, just wait and watch. I'm going to multiply with an I first. Okay? So when you multiply with an I first, let me write first z conjugate thing first. Okay? So this will become 2i and this will become minus 6i square, which is 6. So 6 plus 2i. And here, what will happen? This will give you, so minus 6i square is 6 minus 2i. And this will automatically become minus 20 in that case. Now you check, if you call this as a, if you call this as a, this will be a conjugate. And then this will be a purely real number. Okay? So the story is not over in this step. The story is over in the next step actually. Okay? Anyways, what are you trying to find out? We are trying to find out the equation of the line that is fine. Let us find out its real slope also. What's the real slope? What's the real slope? Real slope is negative real part of A by imaginary part of A. Right? So negative real part of A by imaginary part of A. Real slope would be minus 3. Since it is a real number, that is why we call it as a real number. And complex slope, complex slope is minus A by A conjugate. So minus A by A conjugate. Okay? We need to simplify this. This will also be z1 minus z2, which I have already given to you in that question, by z1 conjugate minus z2 conjugate, please check this out. Okay? Please verify this. This will come out to be the same. I don't want to waste time checking this out because I have a lot of other things to deal with in the next 26 minutes of our class. The next question that we are going to talk about is conditions of parallelism and perpendicularity. Okay? So if you have two lines, L1 and L2, whose complex slopes are known to you, whose complex slopes are known to you, as W1 and W2. W1 and W2 themselves are complex, some remind you. Okay? Then, then prove that if L1 is parallel to L2, then W1 will be equal to W2. What a simple proof, sir. We have done this in our Cartesian coordinate system. Guys, that was a real slope. Okay? Complex slope, I understand. The result is the same, but the process is not the same. Okay? So don't be like, I know, okay, real slope, slopes are equal for parallel lines, so I will say the same for complex slope also. No. They may, they have the same results, but the process is not the same to prove it. Okay? And the next one will be shocking. If L1 is perpendicular to L2, then prove that W1 plus W2 will be equal to 0. Now, this is something different. In case of real slopes, we still have M1 into M2 is minus 1, but when you talk about complex slopes, if two lines are perpendicular or two complex lines are perpendicular, the sum of their complex slopes will be equal to 0. Now, how? Let me prove it. All of you, please just sit and watch. See, let's say I take the first one first. If I have two parallel lines, two parallel complex lines, okay? Let us figure out any two points on it. Let's say I take Z1, Z2 on this line, and let's say I take two points on this Z3, Z4. Okay? Now as for the definition of complex slope, W1 is what? Z1 minus Z2 by Z1 conjugate minus Z2 conjugate. And W2 will be what? Correct? This is known to us? Okay. Now all of you just pay attention over here. If let's say, I mean, I have drawn it like this, but let's say I draw it in the actual Cartesian, sorry, not Cartesian, Argan plane. So let's say my two lines are like this. I'm drawing it like this. One line is like this, another line is like this. Okay? So think as if it is a scenario like this. Z1, Z2, Z3, Z4. Correct? Okay. Now, let us say Z1 minus Z2 is a complex number. So let's say, okay, let me take Z2 minus Z1. It doesn't matter. Nothing will go wrong because just because I've drawn it like this, I have to write Z2 minus Z1. Okay. So let's say there is a complex number Z1 minus Z2, which is making an angle theta with the real Z axis, or possible real Z axis. So can I write it like this? Right? All of you would agree. Can I say similarly, Z4 minus Z3 can be also written in a similar way? Correct? Yes or no? Now many of you would ask me, sir, it could have been theta plus pi also, because it could be in a reverse direction also, right? Agreed with you. It could be both. It could be i theta also. It could be i theta plus pi or theta minus pi also. Okay. Never mind. I will take care of both the scenarios. All of you just be with me. Okay. Now what I'm going to do is I'm going to divide these two. If I'm going to divide these two, I will get Z2 minus Z1 mod, Z4 minus Z3 mod. Now these two will either cancel out. These two will either cancel out or it will give me e to the power plus minus i pi, right? Both the possibilities can be there. So I'm just keeping all the possibilities in the same front. Okay. Now how I will take care of it? I'll square both the sets. Okay. Now whether it is e to the power i0 or whether it is e to the power i pi or whether it is e to the power minus i pi, square of them will always give you a 1. Correct? That means this whole thing will boil down to, this whole thing will boil down to, this, correct? Yes or no? Are you all happy for, happy on this transition? Anybody who is having a problem, please highlight it right now. Okay. Madhav, clear. I know it's complicated. That's why you are here. Correct? All right. Now see what I'm going to do is I'm going to write this in a slightly different way. I'm going to bring this top to the left. Better to write it on the sideways because it's going to end. So this I'm going to do here. Smiley, smiley. Let's move this. So z2 minus z1 square by mod z2 minus z1 square is equal to z4 minus z3 square by mod z4 minus z3 square. Correct? So what I did was I brought this down and I took this up. Right? Why did I do this? You will come to know in some time. Now this denominator, this denominator, can I not write it as z2 minus z1 into z2 minus z1 conjugate? Remember your conjugate properties. Mod z square is z into z conjugate. Correct? Similarly here also, can I write it as z4 minus z3 into z4 minus z3 conjugate? Now one of this will get cancelled with this. One of this will get cancelled with this. Ultimately what do you see? Ultimately you see z2 minus z1. You can actually flip them also. No problem. z1 minus z2 by z1 conjugate minus z2 conjugate equal to, here also let me flip it and write. Not an issue at all. This, actually what do you have written? Actually I have written w1 equal to w2. Hence, isn't it? Wasn't this w1? Check it out. w1 w2 w1 w2 Correct? Okay. Exactly similar way I will also prove the other one also. So first copy this down if you want to. See, complex slope complex slope is a complex number. That is why it is called complex slope. So whatever characteristic a complex number can have the same characteristic a complex slope will have can have. Does that answer your question Abhipya? So a complex slope can be purely real also. Just like a complex number can be a purely real number also. See if z1 equal to z2 God no. It will become fully real. Is it possible that complex slopes are equal but not real or vice versa? I didn't get that question actually. It is beyond my comprehension. Two different lines if they are parallel their complex slope will be equal and the real slope will also be equal as we have already seen. But if two different lines are perpendicular then their real slopes will be multiplied to give you minus 1 whereas their complex slope will add to give you a 0. That is the thing that you want to you should be knowing. Is that what you wanted to know? Good. So now we are going to the next case. Next case is where your lines are perpendicular. So let's say this is our L1 line this is our L2 line and just let us quickly take 2.0 z1 z2 on this. Let's take z3 z4 on this. They are perpendicular. Now see we all know that w1 is z1 minus z2 by z1 conjugate minus z2 conjugate and w2 will be z3 minus z4 by z3 conjugate minus z4 conjugate. Okay. Now all of you please pay attention. Now if let's say I take z1 minus z2. Okay. If I take this as mod z1 minus z2 e to the power i theta then can I say z3 minus z4 will be mod z3 minus z4 e to the power i theta plus minus pi by 2. Now why plus minus? Because it could be having different directions. But don't worry. Even if it is plus pi by 2 or minus pi by 2 I can take care of it. Let us divide first of all. Let's divide 1 by 2. So when you divide 1 by 2 you get z1 by z2 by z3 by z4 minus z4 again the same thing. Okay. But remember when you divide you will either get e to the power plus pi by 2 or minus pi by 2. Correct. Don't worry. I will take care of that. Let me square both the sides. Okay. When you square both the sides please be aware that this term is either plus i or minus i its square will always give you a minus 1. So when you actually square it this is what you will end up seeing. Minus. This bracket was unnecessary actually. So let me remove it. Minus z1 minus z2. Okay. Conjugate. Got it. Yes or no? Now the same method I will apply I will bring this down over here and I will bring this up over here. So this will come down over here. Okay. So I will have z1 minus z2 square by mod z1 minus z2 square and negative z3 minus z4 square by mod z3 minus z4 square. Please be careful. These are round brackets. These are modular symbols. Don't get confused. Right. So in a similar way that I did in the previous case so can I write this as z1 minus z2 z1 conjugate minus z2 conjugate. Similarly this I can write it as z3 minus z4 z3 conjugate minus z4 conjugate. So this gets cancelled with this. This gets cancelled with this. So ultimately what do you see is w1 is equal to negative w2. That means w1 plus w2 is equal to 0. That's true. Okay. So this can be asked as a direct question to you through the form of a comprehension. Through the form of a comprehension. This is a very important result. Again I'm repeating. Not important for JEE main people. So people who are not going to write JEE advance at all despite they qualify. So it is not for them. Okay. Okay. So before we close today's class there is one more important concept that we have to cover is the perpendicular distance. The word perpendicular here was actually not important. Perpendicular distance of a point from a complex line. Sir, are you going to tease the entire coordinate geometry on complex number? No, not really. Okay. Just few related concepts so that if a question comes you should be able to tackle it. Okay. So let's say I give you a complex line whose equation is az conjugate a conjugate z plus b equal to 0. You know why I chose this? Because we had already discussed that the equation of a complex line is of this nature. Okay. b is purely real. Here. And let's say there is a point over here z1. Okay. I want to know what is the perpendicular distance of z1 from this particular line. How will I find this out? Now in the interest of time let me do it because if you try to do it well it will take a lot of time for you. See, let's say I consider this point to be z2. So let's say the foot of the perpendicular bz2. Okay. So let this point be z2. Okay. So can I say when you are finding the value of p you are actually finding out the distance between z1 and z2. Isn't it? So this is what we need to find. Correct? At least the problem statement is clear that we are looking for modulus of z1 minus z2. Now we are completely talking in terms of complex number language. Okay. Though we are using fundamentals that we can borrow from coordinate geometry or vectors we will be basically talking in complex number language only. Let's not deviate from our complex number language. Okay. So this is what we need to find out. Now what we can do? The first thing that I can state over here is that z2 lies on this line. So a z2 conjugate a conjugate z2 plus b should be 0. Let's call this one. Why? Because z2 lies on the given line. Correct? At least this we can say. No doubt about it. Second thing I can say is that the slope of this line let me call this as pm line. Second thing I can say that the complex slope the complex slope of pm plus complex slope of the line should add up to give you 0. Right? Just now we learned w1 plus w2 should be 0. Right? So what is the complex slope of this line? You will say simple sir it is z1 minus z2 by z1 conjugate minus z2 conjugate. Right? So the slope of the pm that you are going you are made over here is this. What is the complex slope of the line minus a by a conjugate that we have already studied? Correct? In other words what you have written over here is z1 minus z2 by z1 conjugate minus z2 conjugate minus a by a conjugate equal to 0. Correct? So this is our second equation let us try to simplify the second equation here. So let's take the LCM so when you take the LCM and you try to simplify this it will become this. Is it fine? Okay? In other words what you have written is a conjugate z1 minus a conjugate z2 minus a z1 conjugate plus a z2 conjugate is equal to 0. Okay? Now all of you pay attention over here very very important. I am not going to manipulate certain things. Sir we know you are very manipulative sir. So this term I will retain over here rest all the terms I am going to send to the other side. So let's write a z1 conjugate plus a conjugate z2 minus a z2 conjugate. Okay? Now what I am going to do is I am going to add certain terms. So I am going to add a z1 conjugate plus b. So here also I will write I will add them a z1 conjugate plus b. Okay? Now this will become 2 a z1 conjugate plus a z2 conjugate minus other way round. So basically I have clubbed these two. This will be a conjugate z2 minus a z2 conjugate plus b. Now please pay attention over here. From this equation then the first equation can I write can I write b as negative a z2 conjugate negative a conjugate z2? Let us substitute this b over here. Let us substitute b over here minus a z2 conjugate minus a z2 conjugate. So this b I am replacing minus a z2 conjugate minus a conjugate z2. Okay? Let us all the terms I will copy as it is. 2 a z1 conjugate a conjugate z2 minus a z2 conjugate. Okay? Now you can see that these two terms will get cancelled off and you will end up getting 2 a z1 conjugate minus 2 a z2 conjugate. Correct? Now this is the last step. We are almost done. Can I say from here I can write it as 2 a z1 conjugate minus z2 conjugate? Correct? So can I say 2 a z1 conjugate minus z2 conjugate is equal to a conjugate z1 a z1 conjugate plus b? So I will divide it by 2a. Correct? Then I will take the modulus on both the sides. Correct? Now this is nothing but mod z1 minus z2 conjugate which is actually mod z1 minus z2 because you know that conjugates modulus and the complex number modulus are same. Can I say this is equal to equal to conjugate of this term sorry, modulus of this term not conjugate by 2 conjugate a so this becomes your distance please make a note of this. Now many people ask me sir we need to remember it. Of course you would not like to do these derivations in your examination hall will you? So please note it down and use it. If you use it well you will remember it. P was never a real number. How do you know that? This is a completely real number. Modulus makes it real. Numerator is the modulus denominator is the modulus. Distance is purely real. Distance is a purely real quantity. Now yes Hariharan has a very good point. It has a similarity with the Cartesian coordinate system but not exactly. The two factor here makes a difference. No it's not the same. That's the point. Two here is coming up. That makes it slightly different. Let's take a quick question on this. First note it down. It's heavy but it's not beyond us. Some quick questions I will take up just to show you that questions have been framed on these concepts. Questions have been framed on these concepts. Where is that question? This one. Now that you know the result I don't think you will take much time to solve this. Find the length of the perpendicular from z0 to this line. Should I put the pole on? Let's put the pole on. You should not take even 5 seconds to answer this. Janta is taking half a minute. It's like somebody is taking a spoon to your mouth and you just turn your face away. You don't even want a spoon feeding. Come on guys it's B. Now there's no point. Let me end the poll. See people are giving wrong answers also. Janta are you sleeping? Okay guys one year from hence you will be filling the form. Right? This one you give 100% right answer. Pole is on. Guys I forgot to take the attendance in the beginning and now that you have left the session will be marked absent. Somebody has marked there also. Come on guys time is not there. One minute, one minute. That too is not even one minute. Few seconds. At the count of 5 I will stop. 5, 4, 3, 2, 1, go. I want to find these three people who have answered A and C. Just now I was barking my lungs out that it is W1 for W2 equal to 0 and still people making mistakes. I am sure it is a slip of the mouse. New term slip of the mouse. Ser... Not really. Mostly they will ask you questions related to complex slopes only. If at all they have to ask. And that too as I told you it is a subject matter of JADvance where they will properly define it because many times these national level exams they don't consider that everybody will know this concept. See guys you are privileged to be very very frank. You people are privileged. Go to places where they don't even know the complex. What is the complex? They don't even know how to add them and they will be writing DE exams. Getting a point. You people are in fact taught much more than you require actually so that you are very confident when you are solving these questions. People from these reputed institutes also they have not even had classes for the last three to four months. Anyways, oh yeah Hariharan we are going to start by normal today. So what we will do is Hariharan will stay back.