If `|z-4 +3i| le 2` then the least and the greatest values of |z| are q

Options

A. 3,7
B. 4,7
C. 3,9
D. 4,5

Answer

To solve the problem, we need to analyze the given inequality involving the complex number \( z \). The inequality is:

\[ |z - (4 - 3i)| \leq 2 \]

This represents all points \( z \) that are within or on the boundary of a circle in the complex plane, centered at the point \( (4, -3) \) with a radius of \( 2 \).

### Step 1: Rewrite the inequality
Let \( z = x + yi \), where \( x \) and \( y \) are real numbers. Then we can rewrite the inequality as:

\[ |(x + yi) - (4 - 3i)| \leq 2 \]

This simplifies to:

\[ |(x - 4) + (y + 3)i| \leq 2 \]

### Step 2: Express the modulus
The modulus can be expressed as:

\[ \sqrt{(x - 4)^2 + (y + 3)^2} \leq 2 \]

### Step 3: Square both sides
Squaring both sides gives:

\[ (x - 4)^2 + (y + 3)^2 \leq 4 \]

This is the equation of a circle centered at \( (4, -3) \) with a radius of \( 2 \).

### Step 4: Find the center and radius
From the equation, we can identify:
- Center \( C(4, -3) \)
- Radius \( r = 2 \)

### Step 5: Calculate the distance from the origin
To find the least and greatest values of \( |z| \), we need to calculate the distance from the origin \( O(0, 0) \) to the center \( C(4, -3) \):

\[ OC = \sqrt{(4 - 0)^2 + (-3 - 0)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 6: Determine least and greatest values of \( |z| \)
The least value of \( |z| \) occurs when we move from the origin towards the center of the circle and then subtract the radius:

\[ \text{Least value} = OC - r = 5 - 2 = 3 \]

The greatest value of \( |z| \) occurs when we move from the origin away from the center of the circle and then add the radius:

\[ \text{Greatest value} = OC + r = 5 + 2 = 7 \]

### Conclusion
Thus, the least and greatest values of \( |z| \) are:

\[ \text{Least value} = 3, \quad \text{Greatest value} = 7 \]

### Final Answer
The values of \( q \) are \( 3 \) and \( 7 \).

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