The circle passing through three distinct points (1,k), (k, 1) and (k, k) passes through the points

Options

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

Answer

To find the fourth point through which the circle passes, we will derive the equation of the circle using the three given points: (1, k), (k, 1), and (k, k). We will then determine the fourth point.

### Step-by-step Solution:

1. **General Equation of Circle**:
The general equation of a circle can be expressed as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where \( g, f, c \) are constants to be determined.

2. **Substituting Points into the Circle Equation**:
We will substitute the three points into the circle equation to create a system of equations.

- For point (1, k):
\[ 1^2 + k^2 + 2g(1) + 2fk + c = 0 \implies 1 + k^2 + 2g + 2fk + c = 0 \quad \text{(Equation 1)} \]

- For point (k, 1):
\[ k^2 + 1^2 + 2g(k) + 2f(1) + c = 0 \implies k^2 + 1 + 2gk + 2f + c = 0 \quad \text{(Equation 2)} \]

- For point (k, k):
\[ k^2 + k^2 + 2g(k) + 2f(k) + c = 0 \implies 2k^2 + 2gk + 2fk + c = 0 \quad \text{(Equation 3)} \]

3. **Simplifying the Equations**:
We now have three equations:
- \( 1 + k^2 + 2g + 2fk + c = 0 \) (1)
- \( k^2 + 1 + 2gk + 2f + c = 0 \) (2)
- \( 2k^2 + 2gk + 2fk + c = 0 \) (3)

4. **Subtracting Equations**:
We will subtract Equation 1 from Equation 2 and Equation 2 from Equation 3 to eliminate \( c \).

- Subtracting (1) from (2):
\[ (k^2 + 1 + 2gk + 2f + c) - (1 + k^2 + 2g + 2fk + c) = 0 \]
This simplifies to:
\[ 2gk + 2f - 2g - 2fk = 0 \implies 2g(k - 1) + 2f(1 - k) = 0 \]
Dividing by 2:
\[ g(k - 1) + f(1 - k) = 0 \quad \text{(Equation 4)} \]

- Subtracting (2) from (3):
\[ (2k^2 + 2gk + 2fk + c) - (k^2 + 1 + 2gk + 2f + c) = 0 \]
This simplifies to:
\[ k^2 + 2fk - 1 - 2f = 0 \implies k^2 + 2f(k - 1) - 1 = 0 \quad \text{(Equation 5)} \]

5. **Solving for \( g \) and \( f \)**:
From Equation 4, we can express \( g \) in terms of \( f \):
\[ g = f \]
Substituting \( g = f \) into Equation 5:
\[ k^2 + 2f(k - 1) - 1 = 0 \]
Rearranging gives:
\[ 2f(k - 1) = 1 - k^2 \implies f = \frac{1 - k^2}{2(k - 1)} \]

6. **Finding \( c \)**:
Substitute \( g \) and \( f \) back into one of the original equations to find \( c \). Using Equation 1:
\[ 1 + k^2 + 2f + 2g + c = 0 \]
Substitute \( g = f \):
\[ 1 + k^2 + 4f + c = 0 \]
Solve for \( c \):
\[ c = -1 - k^2 - 4f \]

7. **Final Circle Equation**:
Substitute \( g \), \( f \), and \( c \) back into the general equation of the circle to find the specific equation.

8. **Finding the Fourth Point**:
The fourth point can be determined by solving the final equation of the circle. After solving, we find that the fourth point is \( (1, 1) \).

### Final Answer:
The fourth point through which the circle passes is \( (1, 1) \).