Division of vectors?? (1 Viewer)

[haboozin]

Hey, i forgot this and now im stuck... too much holidays have made me forget everything....



eg here is a question:

Describe the locus of the complex number w, where z is restricted as indicated:

w = (z - 2i)/(1 - z), |z| = 2

Can someone please write full steps in gaining the answer (preferably using algebra) but more than one method is good too

 

[Captain pi]

Answer

Here is the answer, as I believe:

w = (z – 2i)/(1 – z)

w(1 – z) = (z – 2i) [z =/= 1]

w – wz – z + 2i = 0

z(w + 1) = w + 2i

|z||w + 1| = |w + 2i|

(property of products: moduli equal)
​

2|w + 1| = |w + 2i| (by given |z| = 2)

4|w + 1|² = |w + 2i|²

let w = x + iy​

4((x + 1)² + y²) = x² + (y + 2)²

4(x² + 2x + 1 + y²) = x² + y² + 4y + 4

3x² + 8x + 3y² + 4y = 0

x² + (8/3)x + 16/9 + y² + (4/3)y + 4/9 = 20/9

(x + 4/3)² + (y + 2/3)² = 20/9

This is a circle with centre (–4/3 , –2/3) and radius (2/3)sqrt[5].

I hope that helped.

And so, Captain pi flew off, his task complete. But what will tomorrow bring? Calculus? Number Theory? The Ostrowski-Hadamard Gap Theorem? Only mathematics can tell.

 

[haboozin]

captain pi your are so close,
the correct answer is
locus of w os a circle, radious 2/3(SQRT)5, Center (-4/3,2/3)


this is your mistake:

3x² + 8x + 3y² + 4y = 0

it should be:
3x² + 8x + 3y² - 4y = 0


well done though, thanks for the explination.

And so, Captain pi flew off, his task complete. But what will tomorrow bring? Calculus? Number Theory? The Ostrowski-Hadamard Gap Theorem? Only mathematics can tell.

Wow... that is... so..... nerdy.... I think that reaches new heights of nerdyness. You should write it on your hsc paper HAHAHAHA.

 

[Trev]

And so, Captain pi flew off, his task complete. But what will tomorrow bring? Calculus? Number Theory? The Ostrowski-Hadamard Gap Theorem? Only mathematics can tell.

haha Hugh, your mathematical humour amuses us all

 

[Templar]

And so, Captain pi flew off, his task complete. But what will tomorrow bring? Calculus? Number Theory? The Ostrowski-Hadamard Gap Theorem? Only mathematics can tell.

Meh, I prefer Taniyama-Shimura myself.