The centre of the circle passing through the points `( h,0), (k.0), (0,h), (0,k)` is

Options

A. (a, b)
B. (a/2, b/2)
C. (-a/2, -b/2)
D. (-a, -b)

Answer

To find the center of the circle passing through the points \((h, 0)\), \((k, 0)\), \((0, h)\), and \((0, k)\), we will follow these steps:

### Step 1: Write the general equation of a circle
The general equation of a circle can be expressed as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where \((g, f)\) are related to the center of the circle.

### Step 2: Substitute the first point \((h, 0)\)
Substituting \((h, 0)\) into the circle's equation:
\[ h^2 + 0^2 + 2gh + 2f(0) + c = 0 \]
This simplifies to:
\[ h^2 + 2gh + c = 0 \quad \text{(Equation 1)} \]

### Step 3: Substitute the second point \((k, 0)\)
Substituting \((k, 0)\) into the circle's equation:
\[ k^2 + 0^2 + 2gk + 2f(0) + c = 0 \]
This simplifies to:
\[ k^2 + 2gk + c = 0 \quad \text{(Equation 2)} \]

### Step 4: Subtract Equation 1 from Equation 2
Now, we will subtract Equation 1 from Equation 2:
\[ (k^2 + 2gk + c) - (h^2 + 2gh + c) = 0 \]
This simplifies to:
\[ k^2 - h^2 + 2g(k - h) = 0 \]
Factoring gives:
\[ (k - h)(k + h + 2g) = 0 \]
This implies:
\[ k + h + 2g = 0 \quad \Rightarrow \quad 2g = - (h + k) \quad \Rightarrow \quad g = -\frac{h + k}{2} \]

### Step 5: Substitute the third point \((0, h)\)
Substituting \((0, h)\) into the circle's equation:
\[ 0^2 + h^2 + 2g(0) + 2fh + c = 0 \]
This simplifies to:
\[ h^2 + 2fh + c = 0 \quad \text{(Equation 3)} \]

### Step 6: Substitute the fourth point \((0, k)\)
Substituting \((0, k)\) into the circle's equation:
\[ 0^2 + k^2 + 2g(0) + 2fk + c = 0 \]
This simplifies to:
\[ k^2 + 2fk + c = 0 \quad \text{(Equation 4)} \]

### Step 7: Subtract Equation 3 from Equation 4
Now, we will subtract Equation 3 from Equation 4:
\[ (k^2 + 2fk + c) - (h^2 + 2fh + c) = 0 \]
This simplifies to:
\[ k^2 - h^2 + 2f(k - h) = 0 \]
Factoring gives:
\[ (k - h)(k + h + 2f) = 0 \]
This implies:
\[ k + h + 2f = 0 \quad \Rightarrow \quad 2f = - (h + k) \quad \Rightarrow \quad f = -\frac{h + k}{2} \]

### Step 8: Determine the center of the circle
The center of the circle can be found using the values of \(g\) and \(f\):
\[ \text{Center} = (-g, -f) = \left(\frac{h + k}{2}, \frac{h + k}{2}\right) \]

### Final Answer
Thus, the center of the circle passing through the points \((h, 0)\), \((k, 0)\), \((0, h)\), and \((0, k)\) is:
\[ \left(\frac{h + k}{2}, \frac{h + k}{2}\right) \]