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# P3 Complex number loci. Watch

1. Hello,

u = (6-3i)/(1+2i)

i. For complex number z satisfying arg(z-u) = pi/4. , find the least value of |z|.

ii. For complex numbers satisfying |z-(1+u)| = 1, find the greatest possible value of |z|.

Thanks. ...
2. (Original post by methewthomson)
Hello,

u = (6-3i)/(1+2i)

i. For complex number z satisfying arg(z-u) = pi/4. , find the least value of |z|.

ii. For complex numbers satisfying |z-(1+u)| = 1, find the greatest possible value of |z|.

Thanks. ...
First write u in the form

It should then be clear how to draw the sketches
3. (Original post by Indeterminate)
First write u in the form

It should then be clear how to draw the sketches

Yes, I had some idea that first it should be written in a+bi form. But what is next? That is what I don't get....

Could you please solve it out....perhaps with graphical illustration...
thanks again.
4. (Original post by methewthomson)

Yes, I had some idea that first it should be written in a+bi form. But what is next? That is what I don't get....

Could you please solve it out....perhaps with graphical illustration...
thanks again.
Well, take

and draw the half-line representing

Deduce that

As a further hint, you will need to find the minimum value of the quadratic

and then take the square root of that.
5. (Original post by Indeterminate)
Well, take

and draw the half-line representing

Deduce that

As a further hint, you will need to find the minimum value of the quadratic

and then take the square root of that.
Actually, I'm bad at loci drawings and that is why I could not draw the half line representing . And why do we draw a half line?Could you plzzzz explain?

u = 3i in our case. That is okay. But the next steps I don't understand.
6. (Original post by methewthomson)
Actually, I'm bad at loci drawings and that is why I could not draw the half line representing . And why do we draw a half line?Could you plzzzz explain?

u = 3i in our case. That is okay. But the next steps I don't understand.
u = -3i

Can you draw easier loci?

for example.
7. You would get b+3=a. (1)
if u assume a+ib as z
now you have to minimise a^2 +b^2 =f(x)^2
so put a from 1 and then differentiate to get b then find ur answer.
You would get b+3=a. (1)
if u assume a+ib as z
now you have to minimise a^2 +b^2 =f(x)^2
so put a from 1 and then differentiate to get b then find ur answer.
I can't say I understand what you're trying to do there.

The question certainly doesn't require any differentiation.

The thread title refers to loci. Once the correct loci are sketched a little bit of high school maths is all that is needed.
9. (Original post by BabyMaths)
u = -3i

Can you draw easier loci?

for example.

Yes, I know how to draw that, but just that. I don't know how to draw the complex type.

Can you plz explain the question I put on this thread?
10. (Original post by methewthomson)
Yes, I know how to draw that, but just that. I don't know how to draw the complex type.

Can you plz explain the question I put on this thread?
You're trying to do a sketch for .

Can you sketch ?

Can you then use the relationship between z and w, namely z=w-3i?
11. (Original post by BabyMaths)
You're trying to do a sketch for .

Can you sketch ?

Can you then use the relationship between z and w, namely z=w-3i?
Thank you so very much for replying again.

Yes I can sketch easily . But how will I find the smallest value of w so that z turns out to be smallest, giving smallest modulus???

Could you plz solve the question step by step?Your manner of explanation is really good.

Plz help me as I lose lots of marks on such questions...
thanks....
12. (Original post by methewthomson)
Thank you so very much for replying again.

Yes I can sketch easily . But how will I find the smallest value of w so that z turns out to be smallest, giving smallest modulus???

Could you plz solve the question step by step?Your manner of explanation is really good.

Plz help me as I lose lots of marks on such questions...
thanks....

I'd like to see them before I say any more.
13. (Original post by BabyMaths)

I'd like to see them before I say any more.
Okay, here is the sketch you asked me to draw:
14. (Original post by methewthomson)
...
Try to re-upload your picture, doesn't seem to be working for me.
15. (Original post by Joshmeid)
Try to re-upload your picture, doesn't seem to be working for me.
Now you will be able to see a small picture. Open the picture in a new window to increase its size.
16. (Original post by methewthomson)
Now you will be able to see a small picture. Open the picture in a new window to increase its size.
Ok in that image, you've drawn . (ignoring the -3i and 3 marked on the axis.)

This is the half-line starting from the origin with an angle of .

The argument is written in the form .

In your case we have , which can be re-written as .

We can then represent this on an argand diagram starting at the point (0, -3), with an angle of from the positive x-axis.

Now if you understand that, can you make another sketch of .
17. (Original post by Joshmeid)
Ok in that image, you've drawn .
He's drawn arg(w)=pi/4 as I asked him.

Now, if w=z+3i you have z=w-3i. Just sketch it in 3i below w.
18. (Original post by BabyMaths)
He's drawn arg(w)=pi/4 as I asked him.

Now, if w=z+3i you have z=w-3i. Just sketch it in 3i below w.
I attempted to move away from that to prevent confusion later on when he arrives at transformations from z to w plane but feel free to continue.
19. (Original post by BabyMaths)
He's drawn arg(w)=pi/4 as I asked him.

Now, if w=z+3i you have z=w-3i. Just sketch it in 3i below w.

Should I sketch z in 3i below? At what angle? Parallel to w?
If possible please draw a diagram.
Thanks.
20. (Original post by Joshmeid)
I attempted to move away from that to prevent confusion later on when he arrives at transformations from z to w plane but feel free to continue.
Thank you for all your efforts.

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