Basic questions of complex number's geometry?

In summary, the conversation discusses finding the locus of a complex number z, given different equations and conditions. The first problem has two solutions, one where z=-4 and the other where the first factor is zero. The second problem involves finding the value of an expression with two complex numbers, which can be simplified by expanding and simplifying. The third problem involves finding the locus of z, which is a circle with radius sqrt(3) centered at (-3,0) for all real k values.
  • #1
vkash
318
1

Homework Statement



(z' represent conjugate of complex number z,i is iota =sqrt(-1))
(1)find the locus of z.
|z|2+4z'=0

(2)|z1|=1, |z2|=2, then find the value of |z1-z2|2+|z1+z2|2

(3)z=(k+3)+i[sqrt(3-k2)] for all real k. find locus of z

Homework Equations



|z|2=z.z'

The Attempt at a Solution



(1) |z|2+4z'= |z|2+4z'*z/z
|z|2{1+4/z}=0
it is possible iff z= -4. so it has one solution.

(2) I take z1= cos(a)+i sin(a) and z2=2{cos(b)+i*sin(b)}
after putting these values in the required equation i got 10.

(3) let z=x+iy
x=k+3; y=sqrt(3-k2)
x-3=k; y2=3-k2
squaring x and adding it to y2,
(x-3)2+y2=k2+3-k2
that's circle.

unfortunately all of my answers are incorrect.
I want to know why?
 
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  • #2
For (1), you are missing one solution. You have two factors and the equation is true when either one is zero. [itex]z=-4[/itex] is the solution where the 2nd factor is zero, what is the solution when the 1st factor vanishes?

For (2), it would help to expand as

[itex] | z_1-z_2|^2 + | z_1+z_2|^2 = (z_1-z_2) (\bar{z}_1-\bar{z}_2)+ (z_1+z_2) (\bar{z}_1+\bar{z}_2),[/itex]

then simplify the resulting expression.

For (3), you made a mistake to try to compute (x-3)2+y2. (x-3)2 does not appear in |z|2. Instead, it might help to look at the real and imaginary parts independently as functions of k. Looking at some special points like [itex]k=0,\pm 3[/itex] also helps.
 
  • #3
fzero said:
For (1), you are missing one solution. You have two factors and the equation is true when either one is zero. [itex]z=-4[/itex] is the solution where the 2nd factor is zero, what is the solution when the 1st factor vanishes?

For (2), it would help to expand as

[itex] | z_1-z_2|^2 + | z_1+z_2|^2 = (z_1-z_2) (\bar{z}_1-\bar{z}_2)+ (z_1+z_2) (\bar{z}_1+\bar{z}_2),[/itex]

then simplify the resulting expression.

For (3), you made a mistake to try to compute (x-3)2+y2. (x-3)2 does not appear in |z|2. Instead, it might help to look at the real and imaginary parts independently as functions of k. Looking at some special points like [itex]k=0,\pm 3[/itex] also helps.

(1) Oh it was y foolishness.

(2) even after opening the equation you place i got 2(|z1|2+|z2|2) =2(1+22)=10

(3)I did not understand third answer.

THANKS FOR REPLY first 2 answer really helps me.
In third; 3 and-3 lies on the circle but not satisfy the complex number. so i think it is part of circle for all those for which [itex]k<|\frac{1}{\sqrt{3}}|[/itex]
 
  • #4
For the third one, [itex]z= x+ iy= k+3+ (3- k^2)i[/itex].

x= k+ 3, [itex]y= \sqrt{3- k^2}[/itex]. Solve the first equation for k, the replace k in the second equation with that.
 
  • #5
And what is the locus of z when 3-k2<0? You need to give the points for all real k values.

ehild
 
  • #6
ehild said:
And what is the locus of z when 3-k2<0? You need to give the points for all real k values.

ehild

see this figure
this is circle(in real part)
but why it is different from this figure
 
  • #7
I do not really understand your pictures. In case 3-k2<0 z is real. (The imaginary part is zero.)

ehild
 

1. What are complex numbers?

Complex numbers are numbers that have both real and imaginary components. They are written in the form a + bi, where a and b are real numbers and i is the imaginary unit, equal to the square root of -1.

2. What is the geometric interpretation of complex numbers?

Complex numbers can be represented as points on a complex plane, where the real component is represented on the x-axis and the imaginary component is represented on the y-axis. The distance from the origin to the complex number is called the modulus, and the angle between the positive real axis and the line connecting the origin to the complex number is called the argument.

3. How are complex numbers added and subtracted geometrically?

To add or subtract complex numbers, we simply add or subtract their real and imaginary components separately. Geometrically, this can be seen as translating the complex number on the complex plane by the corresponding real and imaginary components.

4. How are complex numbers multiplied and divided geometrically?

To multiply complex numbers, we use the FOIL (First, Outer, Inner, Last) method, similar to multiplying binomials. Geometrically, this can be seen as scaling and rotating the complex number on the complex plane. To divide complex numbers, we use the conjugate of the denominator to rationalize the fraction, and the result is another complex number. Geometrically, this can be seen as dividing the modulus and subtracting the arguments.

5. What is the geometric significance of the modulus and argument of a complex number?

The modulus of a complex number represents its distance from the origin on the complex plane. The argument represents the angle between the positive real axis and the line connecting the origin to the complex number. Together, they give a complete geometric representation of the complex number on the complex plane.

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