How does one get the form of a circle out of this equation?

  • Thread starter Keshroom
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  • #1
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Homework Statement



Sketch modulus((z+1)/(2z+3))=1 on the complex plane where z=x+iy

Homework Equations





The Attempt at a Solution


I know it is a circle but i need help simplifying the equation into the form of a circle.

i'm stuck at
0= 3x^2 + 3y^2 + 10x + 8

I usually complete the square on these type of problems but it has a 3x^2. What can i do to get it to the form of a circle?
 

Answers and Replies

  • #2
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If you use Latex, this looks neater:
[tex]\left| \frac{z+1}{2z+3} \right| = 1[/tex]
Play around with the absolute value:
[tex]\frac{|z+1|}{|2z+3|} = 1[/tex]
This means that [itex]|z+1|=|2z+3|[/itex].
Now substituting a+bi for z, we get [itex]|(a+1)+bi|=|(2a+3)+2bi|[/itex].
Using the definition of the absolute value of a complex number, we then obtain
[tex]a^2+2a+b^2+1=4a^2+12a+9+4b^2[/tex]
Rearranging yields
[tex]3a^2+10a+3b^2+8=0[/tex]
which you already obtained.
This is a quadratic equation which you can solve to get the imaginary part of z in terms of its real part. Then, plug in values for the real part and get the imaginary part values!
 
  • #3
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i'm stuck at
0= 3x^2 + 3y^2 + 10x + 8

I usually complete the square on these type of problems but it has a 3x^2. What can i do to get it to the form of a circle?
If you want to complete the square but can't because of the 3x2 there, what can you do to remove the 3? Divide by it!
 
  • #4
vela
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Homework Statement



Sketch modulus((z+1)/(2z+3))=1 on the complex plane where z=x+iy

Homework Equations





The Attempt at a Solution


I know it is a circle but i need help simplifying the equation into the form of a circle.

i'm stuck at
0= 3x^2 + 3y^2 + 10x + 8

I usually complete the square on these type of problems but it has a 3x^2. What can i do to get it to the form of a circle?
That's what you want to do. What specifically is it about completing the square that is a problem for you?
 
  • #5
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If you want to complete the square but can't because of the 3x2 there, what can you do to remove the 3? Divide by it!
ofcause!! oh how foolish i am :)

answer:

1/9 = (x+ 5/3)^2 + y^2

thanks guys
 

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