The power of (1,1) with respect to the circle `x^(2)+y^(2)-4x+3y+k=0` is 3, then k

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the value of \( k \) given that the power of the point \( (1, 1) \) with respect to the circle described by the equation \( x^2 + y^2 - 4x + 3y + k = 0 \) is 3, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Power of a Point**:
The power of a point \( (x_0, y_0) \) with respect to a circle defined by the equation \( x^2 + y^2 + Dx + Ey + F = 0 \) is given by the formula:
\[ \text{Power} = x_0^2 + y_0^2 + Dx_0 + Ey_0 + F \]
In our case, the point is \( (1, 1) \) and the circle's equation is \( x^2 + y^2 - 4x + 3y + k = 0 \).

2. **Identify the Coefficients**:
From the equation \( x^2 + y^2 - 4x + 3y + k = 0 \), we can identify:
- \( D = -4 \)
- \( E = 3 \)
- \( F = k \)

3. **Substitute the Point into the Power Formula**:
Substitute \( (x_0, y_0) = (1, 1) \) into the power formula:
\[ \text{Power} = 1^2 + 1^2 - 4(1) + 3(1) + k \]
Simplifying this gives:
\[ \text{Power} = 1 + 1 - 4 + 3 + k = 1 + 1 - 4 + 3 + k = 1 + k \]

4. **Set the Power Equal to 3**:
According to the problem, the power of the point is 3:
\[ 1 + k = 3 \]

5. **Solve for \( k \)**:
Rearranging the equation gives:
\[ k = 3 - 1 = 2 \]

### Conclusion:
Thus, the value of \( k \) is \( 2 \).