If the center of the circle `x^2+y^2+ax+by+2=0` point (4,-8), then a+b=

Options

A. -8
B. -4
C. 4
D. 8

Explanation

To solve the problem, we need to find the values of \( a \) and \( b \) from the equation of the circle given by \( x^2 + y^2 + ax + by + 2 = 0 \), knowing that the center of the circle is at the point \( (4, -8) \).

### Step-by-Step Solution:

1. **Identify the General Equation of a Circle**:
The general equation of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \( (h, k) \) is the center and \( r \) is the radius.

2. **Substitute the Center Coordinates**:
Given the center of the circle is \( (4, -8) \), we can substitute \( h = 4 \) and \( k = -8 \):
\[ (x - 4)^2 + (y + 8)^2 = r^2 \]

3. **Expand the Equation**:
Expanding the left-hand side:
\[ (x - 4)^2 = x^2 - 8x + 16 \]
\[ (y + 8)^2 = y^2 + 16y + 64 \]
Combining these, we have:
\[ x^2 - 8x + 16 + y^2 + 16y + 64 = r^2 \]
Simplifying further gives:
\[ x^2 + y^2 - 8x + 16y + 80 = r^2 \]

4. **Rearranging the Equation**:
Rearranging the equation to set it to zero:
\[ x^2 + y^2 - 8x + 16y + (80 - r^2) = 0 \]

5. **Comparing with the Given Equation**:
The given equation is:
\[ x^2 + y^2 + ax + by + 2 = 0 \]
By comparing coefficients, we can equate:
- For \( x \): \( a = -8 \)
- For \( y \): \( b = 16 \)

6. **Calculate \( a + b \)**:
Now we can find \( a + b \):
\[ a + b = -8 + 16 = 8 \]

### Final Answer:
Thus, the value of \( a + b \) is \( \boxed{8} \).