Equation of the circle on the common chord of the circles `x ^(2) + y ^(2) -ax =0 and x ^(2) + y ^(2) - ay =0` as a diameter is

Options

A. `2x ^(2) + 2y ^(2) -ax -ay =0`
B. `x ^(2) +y ^(2) -ax -ay =0`
C. `2x ^(2) + 2y ^(2) + ax + ay=0`
D. `x ^(2) + y ^(2) -x -y + a=0`

Answer

To find the equation of the circle on the common chord of the circles \( x^2 + y^2 - ax = 0 \) and \( x^2 + y^2 - ay = 0 \) as a diameter, we can follow these steps:

### Step 1: Write the equations of the circles
The given circles can be rewritten as:
1. Circle 1: \( S_1: x^2 + y^2 - ax = 0 \)
2. Circle 2: \( S_2: x^2 + y^2 - ay = 0 \)

### Step 2: Find the equation of the common chord
The equation of the common chord of two circles can be found using the formula:
\[ S_1 - S_2 = 0 \]
Substituting \( S_1 \) and \( S_2 \):
\[ (x^2 + y^2 - ax) - (x^2 + y^2 - ay) = 0 \]
This simplifies to:
\[ -ax + ay = 0 \]
Factoring out \( a \):
\[ a(y - x) = 0 \]
Thus, the equation of the common chord is:
\[ x - y = 0 \quad \text{or} \quad y = x \]

### Step 3: Use the common chord as the diameter of a new circle
The equation of a circle with diameter along the line \( y = x \) can be expressed in terms of its center and radius. The center of this circle lies on the line \( y = x \).

### Step 4: Find the center of the new circle
Let the center of the new circle be \( (h, h) \). The radius of the circle can be determined based on the distance from the center to any point on the line \( y = x \).

### Step 5: Write the equation of the circle
The general equation of a circle with center \( (h, h) \) and radius \( r \) is:
\[ (x - h)^2 + (y - h)^2 = r^2 \]
Since the diameter is along the line \( y = x \), we can use the midpoint of the common chord as the center.

### Step 6: Substitute \( h \)
To find the radius, we can note that the radius will be half the length of the common chord. However, since we are looking for the equation of the circle, we can directly express it as:
\[ (x - \frac{a}{2})^2 + (y - \frac{a}{2})^2 = \left(\frac{a}{2}\right)^2 \]
This leads us to the final equation of the circle.

### Final Equation
The equation of the circle on the common chord of the two given circles as a diameter is:
\[ x^2 + y^2 - ax - ay + \frac{a^2}{2} = 0 \]