Complex numbers q (1 Viewer)

[HazzRat]

In this question, you have to prove |z| = 1 given z is a complex number satisfying |2z - 1| = |z - 2|

Worked solution is above but I have two questions:
1) why does squaring both sides of the equation work to get rid of the modulus sign
2) how did they get from step 5 to step 6 (the one's I circled in red)

Ty in advance

 

[synthesisFR]

It’s just the modulus of a complex
How do u find it? Sqrt(re z sqr+ Imz sqr)
Then if u square that it just gets rid of the root

 

[TankKuno]

View attachment 41023

In this question, you have to prove |z| = 1 given z is a complex number satisfying |2z - 1| = |z - 2|

Worked solution is above but I have two questions:
1) why does squaring both sides of the equation work to get rid of the modulus sign
2) how did they get from step 5 to step 6 (the one's I circled in red)

Ty in advance

because the modulus means the root of the additoon of the real and imaginary components squared, so squaring both sides gets rid of the surd

 

[synthesisFR]

Someone teach me latex pls

 

[TankKuno]

Someone teach me latex pls

mr mai believes in self study smh

 

[HazzRat]

It’s just the modulus of a complex
How do u find it? Sqrt(re z sqr+ Imz sqr)
Then if u square that it just gets rid of the root

because the modulus means the root of the additoon of the real and imaginary components squared, so squaring both sides gets rid of the surd

Ty ty - didn't know a modulus is a surd

 

[carrotsss]

Someone teach me latex pls

some parts are outdated but:

A Short Guide to LaTeX

Hi everyone, With the use of LaTeX becoming more widespread, I've decided to write a short guide to LaTeX. If you have any problems, post in this thread or message me somehow. jetblack2007. The graphical editor supplied to us is excellent, though I think at least a small amount of knowledge of...

boredofstudies.org

 

[HazzRat]

some parts are outdated but:

A Short Guide to LaTeX

Hi everyone, With the use of LaTeX becoming more widespread, I've decided to write a short guide to LaTeX. If you have any problems, post in this thread or message me somehow. jetblack2007. The graphical editor supplied to us is excellent, though I think at least a small amount of knowledge of...

boredofstudies.org

 

[HazzRat]

Another one:
How would you graph this:

 

[carrotsss]

Another one:
How would you graph this:
View attachment 41060

sub in z=x+iy to the LHS and see what happens

 

[HazzRat]

sub in z=x+iy to the LHS and see what happens

ok it's just lol

 

[Luukas.2]

Another one:
How would you graph this:
View attachment 41060

You can use to show it rearranges to , and so is the region .

However, thinking of the expression as a vector, the interpretation of the statement is:

is positioned such that the distance from to is less than 2 units​

Since is located at the reflection of in the real axis, is a purely imaginary vector with its midpoint on the real axis.

So, if the length of this vector must be less than 2 units, then must lie within 1 unit (vertically) from the real axis.

Thus, the region covers all possible solutions for .

---

These kinds of vector interpretations can make more algebraically complicated problems easier.

For example: Find the maximum value of given that and .

The answer is , arising from the four values and that correspond to the four corners of the square that defines the region within which must lie.

 

[HazzRat]

Another one:
I’m just working ahead atm and idk how to do these sorts of questions. For example, how would you do 9(c)?

 

[HazzRat]

View attachment 41244

1) what is point P
2) how did they decide the centre