P3 Complex number loci.

Hello,

Please solve the following...

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

$u = a+bi$

It should then be clear how to draw the sketches

Thank you for the reply.

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

$z=x+iy$

and draw the half-line representing $arg(z-u)$

Deduce that $\tan(\frac{\pi}{4}) =...... = 1$

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

$|z|^2 = ....$

and then take the square root of that.

Thank you again for replying.
Actually, I'm bad at loci drawings and that is why I could not draw the half line representing $arg(z-u)$. 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.

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.

u = -3i

Can you draw easier loci?

$arg(z)=\frac{\pi}{3}$ 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?

You're trying to do a sketch for $arg(z+3i)=\frac{\pi}{4}$.

Can you sketch $arg(w)=\frac{\pi}{4}$?

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 $arg(w)=\frac{\pi}{4}$. 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....

Post your sketch(es).

I'd like to see them before I say any more.

...

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.

Ok in that image, you've drawn $\arg{z} = \frac{\pi}{4}$.

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.

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.