If the circle `x^(2)+y^(2)+2gx+2fy+c=0` is touched by `y=x` at P such that `OP=6sqrt(2)`, then the value of c is

Options

A. 36
B. 144
C. 72
D. none of these

Answer

To solve the problem, we need to find the value of \( c \) in the equation of the circle given by

\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

where the circle touches the line \( y = x \) at a point \( P \) such that the distance from the origin \( O \) to \( P \) is \( 6\sqrt{2} \).

### Step-by-Step Solution:

1. **Identify the center and radius of the circle:**
The center of the circle is given by \( C(-g, -f) \) and the radius \( r \) is given by:
\[ r = \sqrt{g^2 + f^2 - c} \]

2. **Use the distance from the origin to point \( P \):**
We know that the distance from the origin \( O(0, 0) \) to the point \( P \) is \( OP = 6\sqrt{2} \).

3. **Apply the Pythagorean theorem:**
In triangle \( OPC \), where \( O \) is the origin, \( P \) is the point of tangency, and \( C \) is the center of the circle, we can apply the Pythagorean theorem:
\[ OC^2 = OP^2 + CP^2 \]

Here, \( OC \) is the distance from the origin to the center of the circle, which is:
\[ OC = \sqrt{g^2 + f^2} \]

So we have:
\[ g^2 + f^2 = (6\sqrt{2})^2 + r^2 \]

4. **Substituting the radius \( r \):**
Since \( r = \sqrt{g^2 + f^2 - c} \), we can substitute this into the equation:
\[ g^2 + f^2 = 72 + (g^2 + f^2 - c) \]

5. **Simplifying the equation:**
Rearranging gives:
\[ g^2 + f^2 = 72 + g^2 + f^2 - c \]
This simplifies to:
\[ c = 72 \]

### Conclusion:
Thus, the value of \( c \) is \( \boxed{72} \).