For the circle `ax^(2)+y^(2)+bx+dy+2=0` centre is (1,2) then `2b+3d=`

Options

A. `-16`
B. 16
C. 8
D. `-8`

Answer

To solve the problem, we need to find the values of \( b \) and \( d \) from the given equation of the circle and then calculate \( 2b + 3d \).

### Step-by-Step Solution:

1. **Identify the standard form of the circle equation**:
The general equation of a circle is given by:
\[ ax^2 + y^2 + bx + dy + 2 = 0 \]
We can compare this with the standard form:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From this comparison, we can identify that \( a = 1 \), \( g = \frac{b}{2} \), and \( f = \frac{d}{2} \).

2. **Use the center of the circle**:
The center of the circle is given as \( (1, 2) \). From the center coordinates, we have:
\[ h = -g \quad \text{and} \quad k = -f \]
Therefore, we can write:
\[ 1 = -\frac{b}{2} \quad \text{and} \quad 2 = -\frac{d}{2} \]

3. **Solve for \( b \) and \( d \)**:
From the first equation:
\[ 1 = -\frac{b}{2} \implies b = -2 \]
From the second equation:
\[ 2 = -\frac{d}{2} \implies d = -4 \]

4. **Calculate \( 2b + 3d \)**:
Now substitute the values of \( b \) and \( d \) into the expression \( 2b + 3d \):
\[ 2b + 3d = 2(-2) + 3(-4) = -4 - 12 = -16 \]

5. **Final answer**:
Therefore, the value of \( 2b + 3d \) is:
\[ \boxed{-16} \]