A conic is represented by `C-=9x^(2)+4xy+6y^(2)-22x-16y+9=0` Q. The centre 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 a systematic approach using partial differentiation.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The general form of a conic is given by:
\[ Ax^2 + Bxy + Cy^2 + 2Dx + 2Ey + F = 0 \]
For our conic, we have:
- \( A = 9 \)
- \( B = 4 \)
- \( C = 6 \)
- \( D = -11 \) (since \( 2D = -22 \))
- \( E = -8 \) (since \( 2E = -16 \))
- \( F = 9 \)

2. **Differentiate with respect to \( x \)**:
We differentiate the equation with respect to \( x \) while treating \( y \) as a constant:
\[ \frac{\partial C}{\partial x} = 2Ax + By + 2D = 0 \]
Substituting the values:
\[ 2(9)x + 4y - 22 = 0 \implies 18x + 4y - 22 = 0 \]
This simplifies to:
\[ 9x + 2y = 11 \quad \text{(Equation 1)} \]

3. **Differentiate with respect to \( y \)**:
Now, we differentiate the equation with respect to \( y \) while treating \( x \) as a constant:
\[ \frac{\partial C}{\partial y} = Bx + 2Cy + 2E = 0 \]
Substituting the values:
\[ 4x + 2(6)y - 16 = 0 \implies 4x + 12y - 16 = 0 \]
This simplifies to:
\[ x + 3y = 4 \quad \text{(Equation 2)} \]

4. **Solve the system of equations**:
We now have two equations:
- \( 9x + 2y = 11 \) (Equation 1)
- \( x + 3y = 4 \) (Equation 2)

From Equation 2, we can express \( x \) in terms of \( y \):
\[ x = 4 - 3y \]

Substitute \( x \) into Equation 1:
\[ 9(4 - 3y) + 2y = 11 \]
Expanding this gives:
\[ 36 - 27y + 2y = 11 \]
Simplifying:
\[ 36 - 25y = 11 \implies -25y = 11 - 36 \implies -25y = -25 \implies y = 1 \]

Now substituting \( y = 1 \) back into Equation 2 to find \( x \):
\[ x + 3(1) = 4 \implies x + 3 = 4 \implies x = 1 \]

5. **Conclusion**:
The center of the conic \( C \) is at the point \( (1, 1) \).

### Final Answer:
The center of the conic \( C \) is \( (1, 1) \).

---