Key Concepts and Formulas

• Perpendicular Bisector Property: The perpendicular bisector of a chord of a circle passes through the center of the circle.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Relationship between chord length, radius, and distance from center: If $L$ is the length of a chord, $r$ is the radius of the circle, and $d$ is the distance from the center of the circle to the chord, then $(\frac{L}{2})^2 + d^2 = r^2$, or $L = 2\sqrt{r^2 - d^2}$.

Step-by-Step Solution

Step 1: Find the midpoint of the chord formed by (4, 2) and (0, 2)

The circle passes through points A(4, 2) and B(0, 2). We need to find the midpoint of the chord AB. This will help us find the equation of the perpendicular bisector. $M = \left(\frac{4+0}{2}, \frac{2+2}{2}\right) = (2, 2)$ This midpoint is (2, 2).

Step 2: Find the equation of the perpendicular bisector of the chord AB

Since the y-coordinates of A and B are the same, the chord AB is horizontal. Therefore, the perpendicular bisector is a vertical line passing through the midpoint (2, 2). The equation of this line is $x = 2$. The center of the circle must lie on this line.

Step 3: Find the coordinates of the center of the circle

The center of the circle lies on the line $3x + 2y + 2 = 0$ and also on the line $x = 2$. Substituting $x = 2$ into the equation of the line: $3(2) + 2y + 2 = 0$ $6 + 2y + 2 = 0$ $2y = -8$ $y = -4$ So, the center of the circle is (2, -4).

Step 4: Calculate the radius of the circle

The radius is the distance between the center (2, -4) and either of the points (4, 2) or (0, 2). Let's use (4, 2): $r = \sqrt{(4-2)^2 + (2-(-4))^2} = \sqrt{2^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40}$ Thus, $r^2 = 40$.

Step 5: Calculate the distance from the center to the midpoint of the chord whose length we want to find

The midpoint of the chord whose length we want to find is N(1, 2). The distance (d) from the center (2, -4) to N(1, 2) is: $d = \sqrt{(1-2)^2 + (2-(-4))^2} = \sqrt{(-1)^2 + (6)^2} = \sqrt{1 + 36} = \sqrt{37}$ Thus, $d^2 = 37$.

Step 6: Calculate the length of the chord

Using the formula $L = 2\sqrt{r^2 - d^2}$: $L = 2\sqrt{40 - 37} = 2\sqrt{3}$

Common Mistakes & Tips

• Be careful to distinguish between the chord defined by the points (4, 2) and (0, 2), which is used to find the center of the circle, and the chord with midpoint (1, 2), whose length we need to calculate.
• Always double-check your distance calculations to avoid errors.
• Remember that the distance from the center to the chord must be less than the radius for the chord to exist.

Summary: We first found the center of the circle by using the fact that it lies on the perpendicular bisector of the chord defined by (4, 2) and (0, 2), and on the line 3x + 2y + 2 = 0. Then, we calculated the radius of the circle. Finally, we used the distance from the center to the midpoint of the desired chord, along with the radius, to find the length of the chord. The chord length is $2\sqrt{3}$.

Final Answer The final answer is $\boxed{2\sqrt{3}}$, which corresponds to option (C).