Math Problem Statement

Determine complex numbers z such that |z - i|, |iz - i|, and |z - iz| have the same modulus.

Solution

Let us solve the problem:

We are tasked with determining the complex numbers zz such that zi|z - i|, izi|iz - i|, and ziz|z - iz| have the same modulus. Let z=x+yiz = x + yi, where x,yRx, y \in \mathbb{R}.


Step 1: Represent zi|z - i|

The expression ziz - i becomes: zi=(x+yi)i=x+(y1)i.z - i = (x + yi) - i = x + (y-1)i. The modulus of ziz - i is: zi=x2+(y1)2.|z - i| = \sqrt{x^2 + (y-1)^2}.


Step 2: Represent izi|iz - i|

The expression iziiz - i becomes: izi=i(x+yi)i=y+xii=x+(y1)i.iz - i = i(x + yi) - i = -y + xi - i = x + (-y-1)i. The modulus of iziiz - i is: izi=x2+(y1)2.|iz - i| = \sqrt{x^2 + (-y-1)^2}.


Step 3: Represent ziz|z - iz|

The expression zizz - iz becomes: ziz=(x+yi)i(x+yi)=x+yi(y+xi)=x+y+(yx)i.z - iz = (x + yi) - i(x + yi) = x + yi - (-y + xi) = x + y + (y - x)i. The modulus of zizz - iz is: ziz=(x+y)2+(yx)2.|z - iz| = \sqrt{(x + y)^2 + (y - x)^2}.


Step 4: Equate zi=izi|z - i| = |iz - i|

From Steps 1 and 2, we equate the moduli: x2+(y1)2=x2+(y1)2.\sqrt{x^2 + (y-1)^2} = \sqrt{x^2 + (-y-1)^2}. Square both sides: x2+(y1)2=x2+(y1)2.x^2 + (y-1)^2 = x^2 + (-y-1)^2. Simplify: (y1)2=(y1)2.(y-1)^2 = (-y-1)^2. Expand both sides: y22y+1=y2+2y+1.y^2 - 2y + 1 = y^2 + 2y + 1. Simplify: 2y=2y    y=0.-2y = 2y \quad \implies \quad y = 0.


Step 5: Substitute y=0y = 0 into ziz|z - iz|

With y=0y = 0, z=x+0i=xz = x + 0i = x. Let us check if ziz|z - iz| still satisfies the given modulus condition.

ziz=xix=x(1i).z - iz = x - ix = x(1 - i). The modulus of zizz - iz becomes: ziz=x(1i)=x1i=x2.|z - iz| = |x(1 - i)| = |x| \cdot |1 - i| = |x| \cdot \sqrt{2}.

For ziz - i and iziiz - i when y=0y = 0, the modulus becomes x2+1\sqrt{x^2 + 1}. Hence, we equate: x2+1=x2.\sqrt{x^2 + 1} = |x| \cdot \sqrt{2}.


Step 6: Solve for xx

Square both sides: x2+1=2x2.x^2 + 1 = 2x^2. Simplify: x2=1.x^2 = 1. Thus: x=±1.x = \pm 1.


Final Answer:

The complex numbers zz are: z=1orz=1.z = 1 \quad \text{or} \quad z = -1.

Would you like a detailed explanation of any specific step? Let me know!


Related Questions:

  1. What is the geometric interpretation of the modulus condition in this problem?
  2. How do complex transformations like iziz affect the modulus of zz?
  3. Can this problem be solved using polar coordinates for zz?
  4. How would the solution change if the moduli were proportional rather than equal?
  5. What are the implications of zz being real (y=0y = 0) in this problem?

Tip:

When dealing with modulus conditions in complex numbers, always square the moduli to avoid dealing with square roots directly!

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Modulus of Complex Numbers
Algebraic Equations

Formulas

Modulus of a complex number: |a + bi| = sqrt(a^2 + b^2)
Equating moduli: |z - i| = |iz - i| = |z - iz|

Theorems

Properties of Modulus in Complex Numbers
Equality of Modulus in Complex Numbers

Suitable Grade Level

Undergraduate Mathematics or Advanced High School