If |Z-4| + |Z+4|=10 , then the difference between the maximum and the minimum values of |Z| is :

Options

A. 2
B. 3
C. `sqrt(41)-5`
D. 0

Answer

To solve the problem, we start with the given equation:

\[ |Z - 4| + |Z + 4| = 10 \]

Let \( Z = x + iy \), where \( x \) and \( y \) are real numbers. The equation can be interpreted geometrically as the sum of the distances from the point \( Z \) to the points \( 4 \) and \( -4 \) on the real axis being equal to \( 10 \). This describes an ellipse with foci at \( (4, 0) \) and \( (-4, 0) \).

### Step 1: Identify the foci and the major axis length
The foci of the ellipse are at \( (4, 0) \) and \( (-4, 0) \). The distance between the foci is \( 2c \), where \( c = 4 \).

### Step 2: Determine the semi-major axis length
The total distance \( |Z - 4| + |Z + 4| = 10 \) represents \( 2a \) for the ellipse. Thus, we have:

\[ 2a = 10 \implies a = 5 \]

### Step 3: Calculate the semi-minor axis length
Using the relationship \( a^2 = b^2 + c^2 \):

\[ a^2 = 5^2 = 25 \]
\[ c^2 = 4^2 = 16 \]

Substituting these values into the equation gives:

\[ 25 = b^2 + 16 \implies b^2 = 25 - 16 = 9 \implies b = 3 \]

### Step 4: Determine the maximum and minimum values of |Z|
The maximum value of \( |Z| \) occurs at the endpoints of the major axis, which is at \( (5, 0) \):

\[ |Z|_{\text{max}} = 5 \]

The minimum value of \( |Z| \) occurs at the endpoints of the minor axis, which are at \( (0, 3) \) and \( (0, -3) \):

\[ |Z|_{\text{min}} = 3 \]

### Step 5: Calculate the difference between maximum and minimum values
Now, we find the difference between the maximum and minimum values of \( |Z| \):

\[ |Z|_{\text{max}} - |Z|_{\text{min}} = 5 - 3 = 2 \]

Thus, the difference between the maximum and minimum values of \( |Z| \) is:

\[ \boxed{2} \]