If `theta` is the angle subtended by the circle S `-= x ^(2) + y ^(2) + 2 gx +2fy+ c =0` at a point `P (x _(1) , y _(1))` outside the circle and `S_(1) -= x _(1) ^(2) + y _(1) ^(2) + 2 gx _(1) + 2f y _(1) + c,` then `cos theta ` is equal too

Options

A. `(S _(1) + c - g ^(2) - f ^(2))/( S _(1) -c +g ^(2) + f ^(2))`
B. `(S_(1) -c + g ^(2) + f ^(2))/( S _(1) + c - g ^(2) + f ^(2))`
C. `( S _(1) + c + g ^(2) -f ^(2))/( S _(1) - c + g ^(2) -f ^(2))`
D. `(S _(1) -c + g ^(2) + f ^(2))/(S _(1) + c + g ^(2) - f ^(2))`

Answer

To solve the problem, we need to find the value of \( \cos \theta \), where \( \theta \) is the angle subtended by the circle at a point \( P(x_1, y_1) \) outside the circle defined by the equation \( S: x^2 + y^2 + 2gx + 2fy + c = 0 \).

### Step-by-Step Solution:

1. **Identify the Circle's Center and Radius**:
- The general equation of the circle is given by \( S: x^2 + y^2 + 2gx + 2fy + c = 0 \).
- The center of the circle \( C \) is at \( (-g, -f) \).
- The radius \( r \) can be calculated using the formula:
\[ r = \sqrt{g^2 + f^2 - c} \]

2. **Calculate \( S_1 \)**:
- The value \( S_1 \) at the external point \( P(x_1, y_1) \) is given by:
\[ S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c \]

3. **Use the Relationship Between \( \theta \) and the Tangent**:
- The angle \( \theta \) subtended by the circle at point \( P \) can be related to the tangent drawn from point \( P \) to the circle.
- The cosine of the angle \( \theta \) can be expressed in terms of \( S_1 \) and the radius \( r \):
\[ \cos \theta = \frac{S_1 - r^2}{S_1 + r^2} \]

4. **Substituting the Radius**:
- Substitute \( r^2 \) into the equation:
\[ r^2 = g^2 + f^2 - c \]
- Therefore, we have:
\[ \cos \theta = \frac{S_1 - (g^2 + f^2 - c)}{S_1 + (g^2 + f^2 - c)} \]

5. **Final Expression for \( \cos \theta \)**:
- Simplifying the expression:
\[ \cos \theta = \frac{S_1 + c - g^2 - f^2}{S_1 - c + g^2 + f^2} \]

### Conclusion:
Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = \frac{S_1 + c - g^2 - f^2}{S_1 - c + g^2 + f^2} \]