Complex locus Q (1 Viewer)

[middlemarch]

Can somebody please help me with this Question
its from the Fitzpatrick book btw

And here it is:

Describe the locus of:
IzI^2 - 2iz + 2t(1+i) = 0
where z=x+iy and x,y,t are real numbers.

For what values of t can x and y be found so that z satisfies the given eqn?

THX
MIDDLE

 

[KFunk]

Can somebody please help me with this Question
its from the Fitzpatrick book btw

And here it is:

Describe the locus of:
IzI^2 - 2iz + 2t(1+i) = 0
where z=x+iy and x,y,t are real numbers.

For what values of t can x and y be found so that z satisfies the given eqn?

THX
MIDDLE

Use the definition |z| = √(x² + y²) to give you:

x² + y² - 2i(x + iy) + 2t(1+i) = 0

x² + y² - 2ix + 2y + 2t + 2ti = 0


Equating real parts we get:

x² + y² + 2y = -2t ......(1)

Equating imaginary parts we get:

2t = 2x ........(2)


subbing (2) into (1) yeilds:

x² + y² + 2y +2x = 0

(x² + 2x + 1) + (y² + 2x + 1) = 2

(x + 1)² + (y + 1)² = 2 , describing the locus we have a circle centre (-1,-1) with a radius of √2

 

[KFunk]

For the range of the values of t i think this might be a way to go about it:

We know that 2t = 2x ---> t=x

Using our locus definition we know that (-1 - √2) ≤ x ≤ (-1 + √2), i.e. the domain of the circle, so I would assume that -

(-1 - √2) ≤ t ≤ (-1 + √2) however I'm not certain of this. I havn't done a quesiton of exactly this type before.

 

[middlemarch]

Thank you KFunk, thay makes a lot of sense.
: )

middle