Find the inclinations of the axes so that the following equations may represent circles, and find the radius and centre
` x^(2) - xy + y^(2) - 2gx - 2 f y = 0`

Answer

To find the inclinations of the axes so that the given equation represents a circle, and to find the radius and center, we will follow these steps:

### Step 1: Identify the given equation
The given equation is:
\[ x^2 - xy + y^2 - 2gx - 2fy = 0 \]

### Step 2: General form of a circle
The general equation of a circle can be expressed as:
\[ x^2 + y^2 + 2xy \cos \omega - 2hx - 2ky - r^2 = 0 \]
where \( (h, k) \) is the center and \( r \) is the radius.

### Step 3: Compare coefficients
From the given equation and the general form, we can identify:
- Coefficient of \( x^2 \): 1
- Coefficient of \( y^2 \): 1
- Coefficient of \( xy \): -1
- Coefficient of \( x \): -2g
- Coefficient of \( y \): -2f

### Step 4: Set up equations based on coefficients
From the comparison, we have:
1. \( 2 \cos \omega = -1 \)
2. \( -2h = -2g \) → \( h = g \)
3. \( -2k = -2f \) → \( k = f \)

### Step 5: Solve for \( \cos \omega \)
From the first equation:
\[ \cos \omega = -\frac{1}{2} \]
This implies:
\[ \omega = 120^\circ \text{ or } \omega = 240^\circ \]

### Step 6: Find the center \( (h, k) \)
Using the values of \( h \) and \( k \):
- \( h = g \)
- \( k = f \)

Thus, the center of the circle is:
\[ (h, k) = (g, f) \]

### Step 7: Find the radius \( r \)
To find the radius, we use the constant term in the general equation of the circle:
\[ h^2 + k^2 + 2hk \cos \omega - r^2 = 0 \]

Substituting \( h = g \), \( k = f \), and \( \cos \omega = -\frac{1}{2} \):
\[ g^2 + f^2 + 2gf \left(-\frac{1}{2}\right) - r^2 = 0 \]
This simplifies to:
\[ g^2 + f^2 - gf - r^2 = 0 \]
Rearranging gives:
\[ r^2 = g^2 + f^2 - gf \]
Thus, the radius \( r \) is:
\[ r = \sqrt{g^2 + f^2 - gf} \]

### Summary of Results
- The inclination of the axes \( \omega \) is \( 120^\circ \).
- The center of the circle is \( (g, f) \).
- The radius is \( r = \sqrt{g^2 + f^2 - gf} \).