Let P point denoting a complex number z on the complex plane. `i.e. z=Re(z)+i Im(z)," where "i=sqrt(-1)` `if" "Re(z)=x and Im(z)=y,then z=x+iy`.The area of the circle inscribed in the region denoted by `|Re(z)|+|Im(z)|=10` equal to

Options

A. `50pi " sq units " `
B. `100pi " sq units" `
C. `55 " sq units" `
D. `110 " sq units" `

Answer

To solve the problem, we need to find the area of the circle inscribed in the region defined by the equation \( |Re(z)| + |Im(z)| = 10 \).

### Step-by-step Solution:

1. **Understanding the Equation**:
The equation \( |Re(z)| + |Im(z)| = 10 \) represents a diamond (or rhombus) shape in the complex plane, with vertices at the points (10, 0), (0, 10), (-10, 0), and (0, -10).

**Hint**: Recognize that this equation describes a geometric figure in the coordinate plane.

2. **Finding the Area of the Diamond**:
The area of a diamond can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals. In this case, the diagonals are the distances between the points (10, 0) to (-10, 0) and (0, 10) to (0, -10).

- The length of the horizontal diagonal \( d_1 = 10 - (-10) = 20 \).
- The length of the vertical diagonal \( d_2 = 10 - (-10) = 20 \).

**Hint**: Identify the lengths of the diagonals based on the vertices of the diamond.

3. **Calculating the Area**:
Now substituting the values of the diagonals into the area formula:
\[ \text{Area} = \frac{1}{2} \times 20 \times 20 = \frac{1}{2} \times 400 = 200 \]

**Hint**: Make sure to multiply the lengths of the diagonals and then divide by 2 to get the area.

4. **Finding the Radius of the Inscribed Circle**:
The inscribed circle (incircle) of the diamond will touch the midpoints of the sides. The radius \( r \) of the incircle can be calculated using the formula:
\[ r = \frac{\text{Area}}{s} \]
where \( s \) is the semi-perimeter of the diamond. The semi-perimeter \( s \) can be calculated as:
\[ s = \frac{1}{2} \times \text{Perimeter} \]
The perimeter of the diamond is \( 4 \times 10\sqrt{2} \) (since each side length is \( 10\sqrt{2} \)). Thus:
\[ s = 2 \times 10\sqrt{2} = 20\sqrt{2} \]

**Hint**: Remember that the semi-perimeter is half of the total perimeter.

5. **Calculating the Radius**:
Now substituting the area and semi-perimeter into the radius formula:
\[ r = \frac{200}{20\sqrt{2}} = \frac{10}{\sqrt{2}} = 5\sqrt{2} \]

**Hint**: Simplify the fraction carefully to find the radius.

6. **Finding the Area of the Inscribed Circle**:
The area of the inscribed circle is given by:
\[ \text{Area of Circle} = \pi r^2 = \pi (5\sqrt{2})^2 = \pi \times 50 = 50\pi \]

**Hint**: Use the formula for the area of a circle and substitute the radius squared.

### Final Answer:
The area of the circle inscribed in the region denoted by \( |Re(z)| + |Im(z)| = 10 \) is \( 50\pi \) square units.