Key Concepts and Formulas

• Family of Circles: If a circle $S = 0$ is touched by another circle at a point P, the equation of any circle touching S at P can be represented as $S + \lambda T = 0$, where T = 0 is the tangent to S at P, and $\lambda$ is a parameter.
• Tangent to a Circle: The equation of the tangent to the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ at the point $(x_1, y_1)$ is $xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c = 0$.
• Radius of a Circle: For a circle in the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, the radius $r$ is given by $r = \sqrt{g^2 + f^2 - c}$.

Step-by-Step Solution

1. Step 1: Identify the Given Circle and Point of Tangency

We are given the equation of a circle $S_1: x^2 + y^2 + 4x - 6y - 12 = 0$ and the point of tangency $(1, -1)$. We need to find the equation of the tangent to this circle at this point.

2. Step 2: Find the Equation of the Tangent (T = 0)

We use the tangent formula $xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c = 0$ to find the equation of the tangent to the circle $x^2 + y^2 + 4x - 6y - 12 = 0$ at the point $(1, -1)$. From the circle equation, we have $2g = 4$, so $g = 2$; $2f = -6$, so $f = -3$; and $c = -12$. Substituting these values and $(x_1, y_1) = (1, -1)$ into the tangent equation: $x(1) + y(-1) + 2(x+1) - 3(y-1) - 12 = 0$ $x - y + 2x + 2 - 3y + 3 - 12 = 0$ $3x - 4y - 7 = 0$ Thus, the equation of the tangent is $T: 3x - 4y - 7 = 0$.

3. Step 3: Formulate the Equation of Circle C

Using the family of circles concept, the equation of circle C can be represented as $S_1 + \lambda T = 0$, where $S_1 = x^2 + y^2 + 4x - 6y - 12$ and $T = 3x - 4y - 7$. Substituting these values: $(x^2 + y^2 + 4x - 6y - 12) + \lambda(3x - 4y - 7) = 0$ This equation represents all circles touching $S_1$ at $(1, -1)$.

4. Step 4: Determine the Value of $\lambda$

We are given that circle C passes through the point $(4, 0)$. Substituting $x = 4$ and $y = 0$ into the equation from Step 3: $(4^2 + 0^2 + 4(4) - 6(0) - 12) + \lambda(3(4) - 4(0) - 7) = 0$ $(16 + 0 + 16 - 0 - 12) + \lambda(12 - 0 - 7) = 0$ $20 + 5\lambda = 0$ $5\lambda = -20$ $\lambda = -4$

5. Step 5: Find the Complete Equation of Circle C

Substituting $\lambda = -4$ back into the family of circles equation: $(x^2 + y^2 + 4x - 6y - 12) - 4(3x - 4y - 7) = 0$ $x^2 + y^2 + 4x - 6y - 12 - 12x + 16y + 28 = 0$ $x^2 + y^2 - 8x + 10y + 16 = 0$ This is the equation of circle C.

6. Step 6: Calculate the Radius of Circle C

From the equation of circle C: $x^2 + y^2 - 8x + 10y + 16 = 0$, we have $2g = -8$, so $g = -4$; $2f = 10$, so $f = 5$; and $c = 16$. Using the radius formula $r = \sqrt{g^2 + f^2 - c}$: $r = \sqrt{(-4)^2 + (5)^2 - 16}$ $r = \sqrt{16 + 25 - 16}$ $r = \sqrt{25}$ $r = 5$ The radius of circle C is 5.

Common Mistakes & Tips

• Tangent Equation Accuracy: Double-check the calculation of the tangent equation to avoid sign errors or misidentification of g and f.
• Family of Circles Setup: Ensure you are using the correct family of curves formula. Here, it is $S + \lambda T = 0$, where T is the common tangent.
• Algebraic Errors: Be careful during the expansion and simplification steps after substituting the value of λ.

Summary

The problem was solved by recognizing the family of circles concept and formulating the equation of the unknown circle C as $S + \lambda T = 0$, where S is the given circle and T is the common tangent. By using the given point (4, 0) that lies on circle C, we determined the value of λ and subsequently found the equation of circle C. Finally, the radius of circle C was calculated using the standard formula. The final answer is 5, which corresponds to option (A).

Final Answer

The final answer is $\boxed{5}$, which corresponds to option (A).