Key Concepts and Formulas

• The general equation of a circle is given by $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center is $(-g, -f)$ and the radius is $\sqrt{g^2 + f^2 - c}$.
• The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• The area of a kite is half the product of its diagonals. When the diagonals are perpendicular, and if the sum of the squares of the radii equals the square of the distance between the centers ($r_1^2 + r_2^2 = (C_1C_2)^2$), the area of the quadrilateral is $r_1r_2$.

Step-by-Step Solution

Step 1: Find the centers and radii of the two circles.

The equation of the first circle is $x^2 + y^2 - 2x - 2y - 2 = 0$. Comparing with the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we have $2g = -2$, $2f = -2$, and $c = -2$. Therefore, $g = -1$, $f = -1$, and $c = -2$. The center $C_1$ is $(-g, -f) = (1, 1)$, and the radius $r_1 = \sqrt{g^2 + f^2 - c} = \sqrt{(-1)^2 + (-1)^2 - (-2)} = \sqrt{1 + 1 + 2} = \sqrt{4} = 2$.

The equation of the second circle is $x^2 + y^2 - 6x - 6y + 14 = 0$. Comparing with the general form, we have $2g = -6$, $2f = -6$, and $c = 14$. Therefore, $g = -3$, $f = -3$, and $c = 14$. The center $C_2$ is $(-g, -f) = (3, 3)$, and the radius $r_2 = \sqrt{g^2 + f^2 - c} = \sqrt{(-3)^2 + (-3)^2 - 14} = \sqrt{9 + 9 - 14} = \sqrt{4} = 2$.

Step 2: Calculate the distance between the centers $C_1$ and $C_2$.

The distance between $C_1(1, 1)$ and $C_2(3, 3)$ is $C_1C_2 = \sqrt{(3 - 1)^2 + (3 - 1)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$.

Step 3: Check if the condition $r_1^2 + r_2^2 = (C_1C_2)^2$ is satisfied.

We have $r_1^2 = 2^2 = 4$, $r_2^2 = 2^2 = 4$, and $(C_1C_2)^2 = (2\sqrt{2})^2 = 8$. Since $r_1^2 + r_2^2 = 4 + 4 = 8 = (C_1C_2)^2$, the condition is satisfied.

Step 4: Calculate the area of the quadrilateral $PC_1QC_2$.

Since $r_1^2 + r_2^2 = (C_1C_2)^2$, the quadrilateral $PC_1QC_2$ consists of two right-angled triangles $\triangle PC_1C_2$ and $\triangle QC_1C_2$. The area of the quadrilateral is $r_1r_2$. Area $= r_1 \times r_2 = 2 \times 2 = 4$.

Common Mistakes & Tips

• Be careful with the signs when finding the center from the general equation of a circle. Remember that the center is $(-g, -f)$, not $(g, f)$.
• Ensure that you correctly identify the radii and the distance between the centers before applying the area formula.
• Recognizing the quadrilateral as a kite simplifies the area calculation significantly. Also, remembering the condition $r_1^2 + r_2^2 = (C_1C_2)^2$ can lead to a faster solution.

Summary

We first found the centers and radii of the two circles. Then, we calculated the distance between the centers and verified the condition $r_1^2 + r_2^2 = (C_1C_2)^2$. Finally, we used the formula for the area of the quadrilateral (which simplifies to $r_1r_2$ under the given condition) to find the area of the quadrilateral $PC_1QC_2$.

Final Answer The final answer is \boxed{4}, which corresponds to option (A).