Key Concepts and Formulas

• The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, 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 given by the distance formula: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• If a chord of a circle has length $2L$ and the perpendicular distance from the circle's center to the chord is $d$, then the radius $R$ of the circle satisfies the Pythagorean theorem: $R^2 = d^2 + L^2$.

Step-by-Step Solution

Step 1: Determine the Center and Radius of the First Circle

The equation of the first circle is given as $x^2 + y^2 - 10x + 4y + 13 = 0$. We need to find its center and radius.

• Comparing the given equation with the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we get: $2g = -10 \implies g = -5$ $2f = 4 \implies f = 2$ $c = 13$

• Therefore, the center of the first circle ($C_1$) is $(-g, -f) = (5, -2)$.

• The radius of the first circle ($R_1$) is given by: $R_1 = \sqrt{g^2 + f^2 - c} = \sqrt{(-5)^2 + (2)^2 - 13} = \sqrt{25 + 4 - 13} = \sqrt{16} = 4$.

• The diameter of the first circle is $2R_1 = 2 \times 4 = 8$. This diameter is a chord of the second circle C. Therefore, half the length of this chord, $L$, is $L = \frac{8}{2} = 4$.

Why this step? We need the center and radius of the first circle to determine the chord length and the midpoint of the chord, which is the center of the first circle. This midpoint will be used later to calculate the perpendicular distance from the center of the second circle to the chord.

Step 2: Determine the Center of Circle C

The center of circle C is the point of intersection of the lines $2x + 3y = 12$ and $3x - 2y = 5$. We solve this system of equations:

1. $2x + 3y = 12$
2. $3x - 2y = 5$

• Multiply equation (1) by 2 and equation (2) by 3 to eliminate $y$: $2 \times (2x + 3y = 12) \implies 4x + 6y = 24$ $3 \times (3x - 2y = 5) \implies 9x - 6y = 15$

• Add the two new equations: $(4x + 6y) + (9x - 6y) = 24 + 15$ $13x = 39$ $x = \frac{39}{13} = 3$

• Substitute $x = 3$ into equation (1): $2(3) + 3y = 12$ $6 + 3y = 12$ $3y = 6$ $y = 2$

• Therefore, the center of circle C ($C_C$) is $(3, 2)$.

Why this step? The center of the second circle is needed to calculate the perpendicular distance to the chord, which is essential for finding the radius of circle C.

Step 3: Calculate the Perpendicular Distance from the Center of Circle C to the Chord

The chord of circle C is the diameter of the first circle. The midpoint of this diameter is the center of the first circle, $C_1 = (5, -2)$. The perpendicular distance $d$ from the center of circle C ($C_C$) to the chord is the distance between $C_C$ and $C_1$.

• We have $C_C = (3, 2)$ and $C_1 = (5, -2)$.

• Using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$: $d = \sqrt{(5 - 3)^2 + (-2 - 2)^2} = \sqrt{(2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20}$

Why this step? This distance is one leg of the right triangle formed by the radius of the second circle, half the chord length, and the perpendicular distance.

Step 4: Calculate the Radius of Circle C

Let $R_C$ be the radius of circle C. We know:

• Half the length of the chord, $L = 4$ (from Step 1).
• The perpendicular distance from $C_C$ to the chord, $d = \sqrt{20}$ (from Step 3).

Applying the Pythagorean theorem: $R_C^2 = d^2 + L^2$ $R_C^2 = (\sqrt{20})^2 + (4)^2 = 20 + 16 = 36$ $R_C = \sqrt{36} = 6$

Why this step? We use the calculated values of half the chord length and the perpendicular distance to find the radius of the second circle using the Pythagorean theorem.

Common Mistakes & Tips

• Be careful with signs when determining the center of the circle from its general equation.
• Remember that the perpendicular from the center of a circle to a chord bisects the chord.
• Ensure the Pythagorean theorem is set up correctly, with the radius of the larger circle as the hypotenuse.

Summary

We first found the center and radius of the given circle. This allowed us to determine the length of the chord in the second circle and the midpoint of this chord, which is the center of the first circle. Next, we found the center of the second circle by solving the system of linear equations. We then calculated the perpendicular distance from the center of the second circle to the chord (diameter of the first circle). Finally, we used the Pythagorean theorem to find the radius of the second circle, which is 6.

Final Answer The final answer is \boxed{6}, which corresponds to option (C).