Find the locus of a complex number `z=x +yi`, satisfying the relation `|z +i|=|z+ 2|`. Illustrate the locus of z in the Argand plane

Answer

To find the locus of the complex number \( z = x + yi \) satisfying the relation \( |z + i| = |z + 2| \), we will follow these steps:

### Step 1: Write the given condition in terms of \( z \)
We start with the equation:
\[ |z + i| = |z + 2| \]
Substituting \( z = x + yi \), we have:
\[ |(x + yi) + i| = |(x + yi) + 2| \]
This simplifies to:
\[ |x + (y + 1)i| = |(x + 2) + yi| \]

### Step 2: Express the moduli
The modulus of a complex number \( a + bi \) is given by \( \sqrt{a^2 + b^2} \). Thus, we have:
\[ \sqrt{x^2 + (y + 1)^2} = \sqrt{(x + 2)^2 + y^2} \]

### Step 3: Square both sides
To eliminate the square roots, we square both sides:
\[ x^2 + (y + 1)^2 = (x + 2)^2 + y^2 \]

### Step 4: Expand both sides
Expanding both sides gives:
\[ x^2 + (y^2 + 2y + 1) = (x^2 + 4x + 4) + y^2 \]
This simplifies to:
\[ x^2 + y^2 + 2y + 1 = x^2 + 4x + 4 + y^2 \]

### Step 5: Cancel common terms
We can cancel \( x^2 \) and \( y^2 \) from both sides:
\[ 2y + 1 = 4x + 4 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 4x - 2y + 3 = 0 \]

### Step 7: Identify the locus
The equation \( 4x - 2y + 3 = 0 \) represents a straight line in the Cartesian plane.

### Step 8: Illustrate the locus in the Argand plane
To illustrate this line, we can find the intercepts:
- When \( x = 0 \):
\[ 4(0) - 2y + 3 = 0 \implies -2y + 3 = 0 \implies y = \frac{3}{2} \]
So, the point is \( (0, \frac{3}{2}) \).

- When \( y = 0 \):
\[ 4x - 2(0) + 3 = 0 \implies 4x + 3 = 0 \implies x = -\frac{3}{4} \]
So, the point is \( (-\frac{3}{4}, 0) \).

Now, we can plot these points on the Argand plane (complex plane) and draw the line through them.

### Final Locus
The locus of the complex number \( z \) is a straight line given by the equation:
\[ 4x - 2y + 3 = 0 \]