IIT JEE Complex Numbers (part 2) 2010 IIT JEE Paper 1 Problem 39 Complex Numbers (part 2)
IIT JEE Complex Numbers (part 2)
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- In the last video we saw that choice A is true. Now lets see if B, C, and D are also true. So choice B they're telling us that the argument, let me rewrite it over here in another color.
- That choice B, the argument of z - z^1 is equal to the argument of z- z^2. Now lets think about this
- a little bit. We already did some work we simplified what (z - z^1) is equal to. It's equal to T times
- Z^2 - Z^1. So this is equivalent to saying that the argument of T times Z^2 minus Z^1, I'm just leveraging
- some of the algebra that we already did. Because this thing is the same thing as what's inside of the
- absolute value sign which is the same thing as this thing over here. So the argument in T time Z^2 minus
- z^1 is equal to, so I'm just rewriting statement B. Statement B is saying that this should be equal to
- the argument of... now what's (z minus z^2? z minus z^2 we've figured out was this thing right over here.
- Actually it's this thing right over here. It's equal to this thing right over here. And we could rewrite
- this thing as being equal to... I'm running out of real estate here. But we could rewrite this thing
- right here as being equal to 1 minus T, I'm just factoring out the (1 minus t) times (z^1 minus z^2). (z^1
- minus z^2). Or, so we could get it in the same form here let's multiply this times -1 and this times -1
- So I'm multiplying by -1 twice so I'm not changing the number. So this is going to be equal to, or the
- statement is claiming this is equal to the argument of T minus 1. Let me write it a little bit neater
- than that. So the claim is that this should be equal to the argument of (t minus 1) times (z^2 minus z^1).
- So lets think about this and just remember all I did is I multiplied this by negative one and this by
- negative one if I multiply 2 things by negative one it's equivalent to multiplying by negative one twice.
- Which is just multiplying it by one so I was able to swap both of these. So, is this true? Is it true
- that the argument of T times (z^2 minus z^1), is it true that that's the same thing as the argument of
- t minus 1 times (z^2 minus z^1). So let's think about this a little bit. Let's draw an Argand Diagram.
- Here... that's the imaginary axis, this is the real axis. And, let's draw the vectored Z^2 minus Z^1
- so let's say that this here is the vector and lets draw the vector Z^2 minus Z^1
- now what would T times Z^2 minus Z^1 look like? Well T, we learned, is between zero and 1. So, it's going
- to be a scaled down version of Z^2 minus Z^1. So this right here, so who knows what it is? This right
- here would be T times Z^2 minus Z^1. Now what would T minus 1 times Z^2 minus Z^1 look like? Well T minus
- 1 if you remember T is between zero and 1. So T is less than 1. So T minus one, this right here is going
- to be a negative number. It's going to be a negative number. So we're going to be scaling it by a negative
- number. So we're going to be scaling Z^2 minus Z^1 by a negative number. So this thing right over here.
- (t-1) times (z^2 minus z^1) is going to look like this. This is (t minus 1) times (z^2 minus z^1). Now
- the arguments are just the angles between each of these numbers and the real axis so the argument for
- this thing right over here is going to be this angle is going to be 5 right over so I could call that
- 5. But what's the argument for this thing? It's going to be that plus pi or plus 180 degrees. We have
- to go all the way around. So they are not the same angle. Whatever number this is, right over here,
- this number is going to be plus pi, or you can even go minus pi. But they're definitely not
- going to be the same angle. So we can cross out, choice B as an option. Now that we're in this mode
- let's see if we can tackle choice D because it looks very similar. And then I'll probably do
- choice C in the next video because I don't want to spend too much time in each video. So what is choice
- D telling us? Let me write it down here. Choice D is telling us... so we don't know, we have to see
- if it's true that the Argument of (z minus z^1) is equal to the argument of (z^2 minus z^1)
- So let's think a little bit about this right now. So, once again, we've already figured out what (z minus
- z^1) is equal to. It's equal to this business over here. So this statement is equivalent
- to the statement that the argument of (z minus z^1) which is (t times z^2 minus z^1) leveraging the algebra
- from the last video. (t time z^2 minus z^1). Is this equal to the argument of (z^2 minus z^1)? Well once
- again let's draw our Argand Diagram. Actually we could leverage the same Argand Diagram in green. We
- have (z^2 minus z^1) its argument would be this angle right over here. It would be this angle five. This
- is (z^2 minus z^1) That's that angle. (t times z^2 minus z^1) is just going to be the scaled down version
- of it what we have in orange. Now clearly T is positive so it's not pointing in another direction
- this is just a scaled down version of this. This vector and this vector, or this complex number
- and this complex number are going to point in the same direction. So their angles are the same. This
- orange vector is this here or this orange complex number is this right over here. This is
- (t times z^2 minus z^1) and the green one, just to be clear is (z^2 minus z^1) is that. They're clearly
- in the same direction, they clearly have the same argument. So choice D is another correct choice.
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