If `|z-3i|=|z+3i|`, then find the locus of z.

Answer

To find the locus of the complex number \( z \) such that \( |z - 3i| = |z + 3i| \), we can follow these steps:

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part of the complex number.

### Step 2: Rewrite the equation using the definition of modulus
The given equation can be rewritten as:
\[ |z - 3i| = |z + 3i| \]
Substituting \( z = x + iy \):
\[ |x + iy - 3i| = |x + iy + 3i| \]
This simplifies to:
\[ |x + i(y - 3)| = |x + i(y + 3)| \]

### Step 3: Use the modulus formula
The modulus of a complex number \( a + ib \) is given by \( \sqrt{a^2 + b^2} \). Therefore, we can write:
\[ \sqrt{x^2 + (y - 3)^2} = \sqrt{x^2 + (y + 3)^2} \]

### Step 4: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ x^2 + (y - 3)^2 = x^2 + (y + 3)^2 \]

### Step 5: Simplify the equation
Cancelling \( x^2 \) from both sides, we have:
\[ (y - 3)^2 = (y + 3)^2 \]
Expanding both sides:
\[ y^2 - 6y + 9 = y^2 + 6y + 9 \]

### Step 6: Cancel common terms
Cancelling \( y^2 \) and \( 9 \) from both sides results in:
\[ -6y = 6y \]

### Step 7: Solve for \( y \)
Bringing all terms involving \( y \) to one side gives:
\[ -6y - 6y = 0 \implies -12y = 0 \implies y = 0 \]

### Conclusion: Identify the locus
The equation \( y = 0 \) represents the real axis in the complex plane. Therefore, the locus of \( z \) is the real axis.

### Final Answer
The locus of \( z \) is the real axis.

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