If z be a complex number satisfying `|z-4+8i|=4`, then the least and the greatest value of `|z+2|` are respectively (where `i=sqrt(-i)`)

Options

A. 7 and 16
B. 8 and 17
C. 6 and 14
D. 5 and 13

Answer

To solve the problem, we need to find the least and greatest values of \( |z + 2| \) given that \( |z - 4 + 8i| = 4 \).

### Step 1: Understand the given condition

The condition \( |z - 4 + 8i| = 4 \) represents a circle in the complex plane centered at the point \( (4, -8) \) with a radius of \( 4 \).

### Step 2: Express \( z \)

Let \( z = x + yi \), where \( x \) and \( y \) are real numbers. The condition can be rewritten as:

\[ |(x - 4) + (y + 8)i| = 4 \]

This implies:

\[ \sqrt{(x - 4)^2 + (y + 8)^2} = 4 \]

Squaring both sides gives:

\[ (x - 4)^2 + (y + 8)^2 = 16 \]

### Step 3: Find the expression for \( |z + 2| \)

We want to find \( |z + 2| \):

\[ |z + 2| = |(x + 2) + yi| = \sqrt{(x + 2)^2 + y^2} \]

### Step 4: Analyze the geometric situation

We need to find the minimum and maximum values of \( |z + 2| \) as \( z \) varies on the circle defined by \( (x - 4)^2 + (y + 8)^2 = 16 \).

The center of the circle is \( (4, -8) \) and the radius is \( 4 \). The point \( (-2, 0) \) corresponds to \( z + 2 \) being evaluated.

### Step 5: Calculate the distance from the center to the point

First, calculate the distance from the center of the circle \( (4, -8) \) to the point \( (-2, 0) \):

\[ d = \sqrt{(4 - (-2))^2 + (-8 - 0)^2} = \sqrt{(4 + 2)^2 + (-8)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

### Step 6: Determine the minimum and maximum values

The minimum value of \( |z + 2| \) occurs when \( z \) is closest to \( (-2, 0) \), which is at a distance of \( 10 - 4 = 6 \) (subtracting the radius of the circle).

The maximum value of \( |z + 2| \) occurs when \( z \) is farthest from \( (-2, 0) \), which is at a distance of \( 10 + 4 = 14 \) (adding the radius of the circle).

### Conclusion

Thus, the least and greatest values of \( |z + 2| \) are:

- Least value: \( 6 \)
- Greatest value: \( 14 \)

### Final Answer

The least and greatest values of \( |z + 2| \) are \( 6 \) and \( 14 \), respectively.

---