If z `in` C, which of the following relation(s) represents a circle on an Argand diagram? (where,`i=sqrt(-1)`)

Options

A. `abs(z-1)+abs(z+1)=3`
B. `abs(z-3)=2`
C. `abs(z-2+i)=7/3`
D. `(z-3+i)(bar(z)-3-i)=5`

Answer

To determine which of the given relations represents a circle on the Argand diagram, we need to analyze each option based on the standard form of the equation of a circle in the complex plane. The standard form is given by:

\[ |z - \alpha| = r \]

where \( \alpha \) is the center of the circle and \( r \) is the radius.

Let's analyze each option step by step.

### Step 1: Analyze Option 1
**Given:** \( |z - 1| + |z + 1| = 3 \)

This equation involves the sum of distances from two points (1, 0) and (-1, 0). This represents an ellipse, not a circle, because it involves two fixed points.

**Conclusion:** Option 1 does not represent a circle.

### Step 2: Analyze Option 2
**Given:** \( |z - 3| = 2 \)

Here, we can see that this is in the form \( |z - \alpha| = r \), where \( \alpha = 3 \) (which corresponds to the point (3, 0) on the Argand plane) and \( r = 2 \).

**Conclusion:** Option 2 represents a circle with center (3, 0) and radius 2.

### Step 3: Analyze Option 3
**Given:** \( |z - (2 - i)| = \frac{7}{3} \)

This can be rewritten as \( |z - (2 + (-1)i)| = \frac{7}{3} \). Here, the center is (2, -1) and the radius is \( \frac{7}{3} \).

**Conclusion:** Option 3 represents a circle with center (2, -1) and radius \( \frac{7}{3} \).

### Step 4: Analyze Option 4
**Given:** \( |z - 3 + i| \cdot |z - 3 - i| = 5 \)

To analyze this, we can substitute \( z = x + iy \) and \( z^* = x - iy \). This leads to:

\[ |(x - 3) + (y + 1)i| \cdot |(x - 3) + (y - 1)i| = 5 \]

This can be simplified to:

\[ (x - 3)^2 + (y + 1)^2 = 5 \]

This is indeed the equation of a circle with center (3, -1) and radius \( \sqrt{5} \).

**Conclusion:** Option 4 represents a circle.

### Final Conclusion
The options that represent circles on the Argand diagram are:
- Option 2: \( |z - 3| = 2 \)
- Option 3: \( |z - (2 - i)| = \frac{7}{3} \)
- Option 4: \( |z - 3 + i| \cdot |z - 3 - i| = 5 \)