The points representing complex number z for which `|z+2+ 3i| = 5` lie on the locus given by

Options

A. circle
B. ellipse
C. straight line
D. None of these

Answer

To find the locus of the complex number \( z \) for which \( |z + 2 + 3i| = 5 \), we can follow these steps:

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = x + yi \), where \( x \) and \( y \) are real numbers.

### Step 2: Rewrite the equation
Substituting \( z \) into the equation gives:
\[ |z + 2 + 3i| = |(x + yi) + 2 + 3i| = |(x + 2) + (y + 3)i| \]

### Step 3: Use the modulus definition
The modulus of a complex number \( a + bi \) is given by \( \sqrt{a^2 + b^2} \). Therefore, we have:
\[ |(x + 2) + (y + 3)i| = \sqrt{(x + 2)^2 + (y + 3)^2} \]

### Step 4: Set up the equation
According to the problem, this modulus equals 5:
\[ \sqrt{(x + 2)^2 + (y + 3)^2} = 5 \]

### Step 5: Square both sides
To eliminate the square root, we square both sides:
\[ (x + 2)^2 + (y + 3)^2 = 25 \]

### Step 6: Identify the locus
The equation \( (x + 2)^2 + (y + 3)^2 = 25 \) represents a circle in the Cartesian plane.

### Step 7: Determine the center and radius
From the standard form of the circle \( (x - h)^2 + (y - k)^2 = r^2 \):
- The center of the circle is \( (-2, -3) \).
- The radius \( r \) is \( 5 \) (since \( r^2 = 25 \)).

### Conclusion
Thus, the locus of the points representing the complex number \( z \) is a circle with center at \( (-2, -3) \) and radius \( 5 \).

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