another complex question (locus) (1 Viewer)

[c0okies]

hi i have quite a few complex questions which i have absolutely no idea what to do:

1) if z1= 24+7i and |z2|=6. Find the greatest and smallest values of |z1+z2|

2) On an Argand diagram the points A and B represent the numbers z1 and z2 respectively. I is the point (1,0). D is the point such that triangle OID is similar to triangle OBA. Show that D represents z1/z2.

3) If |z1+z2| = |z1-z2|, find the possible values of arg(z1/z2).

4) On Argand Diagram the point P and Q represent the numbers z1 and z2 respectively. OPQ is an equilateral triangle. SHow that z1^2 +z2^2 =z1z2.

thanks for any help

 

[pLuvia]

1.
z1 = 24+7i
|z2| = 6

.: |z1+z2|<|z1| + |z2|
< 25 + 6
< 31

Greated value when z1=kz2, where k>0

|z1+z2|>|z1|-|z2|
> 25 - 6 =19

Minimum value when z1=kz2 where k>0

2.

If you draw it out

D = z1/z2

Let z1 = cis #
Let z2 = cis @

Since triangle OID ||| triangle OBA

OA/OD = OB/OI
z1/z3 = z2/1
z1/z3 = z2
z3 = z1/z2
.: as req'd

3.

If |z1+z2| = |z1-z2|
.: diagonals are equal ( if you draw it out)

OABC is a rectangle or a square

.: arg(z1/z2) = argz1 - argz2 = +/- pi/2

 

[pLuvia]

4.

Let z₁ = z₂ cis pi/3
z²₁ = z²₂ cis 2pi/3

LHS = z²₁ + z²₂
= z²₂cis 2pi/3 + z²₂
= z²₂(cis 2pi/3 +1)
= z²₂(1 + cos 2pi/3 + isin 2pi/3)
= z²₂(1/2 + i(root 3)/2)

RHS = z₁. z₂
= z₂ cis pi/3 . z₂
= z²₂ cis pi/3
= z²₂(1/2 + i(root3)/2)
= LHS as req'd

 

[c0okies]

1.
z1 = 24+7i
|z2| = 6

.: |z1+z2|<|z1| + |z2|
< 25 + 6
< 31

Greated value when z1=kz2, where k>0

|z1+z2|>|z1|-|z2|
> 25 - 6 =19

Minimum value when z1=kz2 where k>0

i dont get it; why must |z1 + z2|< |z1| + |z2| and how did u get 25 +6 ?
sorry if im getting difficult =S

ok i get it =] thanks

 

[pLuvia]

no problem

due to triangle inequality, this is from cambridge so I've done it before

|z₁| is the modulus of z₁ right? so |z₁| = (24²+7²)^1/2
= 25

and we already know |z₂| = 6