A conic is represented by `C-=9x^(2)+4xy+6y^(2)-22x-16y+9=0` Q. The center of conic C is

Options

A. (0,0)
B. (1,0)
C. (0,1)
D. (1,1)

Answer

To find the center of the conic represented by the equation \( C = 9x^2 + 4xy + 6y^2 - 22x - 16y + 9 = 0 \), we will follow these steps:

### Step 1: Write down the equation of the conic
The given equation of the conic is:
\[ 9x^2 + 4xy + 6y^2 - 22x - 16y + 9 = 0 \]

### Step 2: Compute the partial derivatives
To find the center, we need to compute the partial derivatives of \( C \) with respect to \( x \) and \( y \), and set them to zero.

1. **Partial derivative with respect to \( x \)**:
\[ \frac{\partial C}{\partial x} = 18x + 4y - 22 \]
Setting this equal to zero gives us:
\[ 18x + 4y - 22 = 0 \quad \text{(Equation 1)} \]

2. **Partial derivative with respect to \( y \)**:
\[ \frac{\partial C}{\partial y} = 4x + 12y - 16 \]
Setting this equal to zero gives us:
\[ 4x + 12y - 16 = 0 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
We now have a system of two equations:
1. \( 18x + 4y - 22 = 0 \)
2. \( 4x + 12y - 16 = 0 \)

#### Step 3.1: Solve Equation 1 for \( y \)
From Equation 1:
\[ 4y = 22 - 18x \implies y = \frac{22 - 18x}{4} = \frac{11 - 9x}{2} \]

#### Step 3.2: Substitute \( y \) in Equation 2
Substituting \( y \) into Equation 2:
\[ 4x + 12\left(\frac{11 - 9x}{2}\right) - 16 = 0 \]
Simplifying:
\[ 4x + 6(11 - 9x) - 16 = 0 \]
\[ 4x + 66 - 54x - 16 = 0 \]
\[ -50x + 50 = 0 \]
\[ x = 1 \]

#### Step 3.3: Substitute \( x \) back to find \( y \)
Now substitute \( x = 1 \) back into Equation 1:
\[ 18(1) + 4y - 22 = 0 \]
\[ 18 + 4y - 22 = 0 \]
\[ 4y - 4 = 0 \implies y = 1 \]

### Step 4: Conclusion
The center of the conic is:
\[ (x, y) = (1, 1) \]