If the area of the circle `x^(2)+y^(2)+4x+2y+k=0` is `5pi` square cms then k=

Options

A. `-20`
B. 20
C. `+-20`
D. 0

Answer

To solve the problem, we need to find the value of \( k \) in the equation of the circle given by:

\[ x^2 + y^2 + 4x + 2y + k = 0 \]

We know that the area of the circle is \( 5\pi \) square centimeters.

### Step-by-Step Solution:

1. **Rearranging the Circle Equation**:
Start with the given equation of the circle:
\[ x^2 + y^2 + 4x + 2y + k = 0 \]
Rearranging gives:
\[ x^2 + 4x + y^2 + 2y = -k \]

2. **Completing the Square**:
We will complete the square for the \( x \) and \( y \) terms.
- For \( x^2 + 4x \):
\[ x^2 + 4x = (x + 2)^2 - 4 \]
- For \( y^2 + 2y \):
\[ y^2 + 2y = (y + 1)^2 - 1 \]
Substituting these back into the equation gives:
\[ (x + 2)^2 - 4 + (y + 1)^2 - 1 = -k \]
Simplifying this, we have:
\[ (x + 2)^2 + (y + 1)^2 - 5 = -k \]
Thus,
\[ (x + 2)^2 + (y + 1)^2 = k + 5 \]

3. **Identifying the Radius**:
The equation of the circle can be compared to the standard form:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
From our equation, we see that:
\[ r^2 = k + 5 \]

4. **Using the Area of the Circle**:
The area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
Given that the area is \( 5\pi \):
\[ \pi r^2 = 5\pi \]
Dividing both sides by \( \pi \):
\[ r^2 = 5 \]

5. **Setting the Two Expressions for \( r^2 \) Equal**:
We have two expressions for \( r^2 \):
\[ k + 5 = 5 \]
Solving for \( k \):
\[ k = 5 - 5 = 0 \]

### Final Answer:
Thus, the value of \( k \) is:
\[ \boxed{0} \]