Key Concepts and Formulas

• The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, with center $(-g, -f)$ and radius $\sqrt{g^2 + f^2 - c}$.
• The standard equation of a circle is $(x-h)^2 + (y-k)^2 = r^2$, with center $(h, k)$ and radius $r$.
• If a circle has radius $R$ and a chord of length $L$, and the perpendicular distance from the center of the circle to the chord is $d$, then $R^2 = d^2 + \left(\frac{L}{2}\right)^2$.

Step-by-Step Solution

Step 1: Analyze the first circle ($C_1$)

The equation of the first circle is $x^2 + y^2 - 2\sqrt{2}x - 6\sqrt{2}y + 14 = 0$. We want to find its center and radius.

Comparing this to the general equation $x^2 + y^2 + 2gx + 2fy + c = 0$, we have: $2g = -2\sqrt{2} \implies g = -\sqrt{2}$ $2f = -6\sqrt{2} \implies f = -3\sqrt{2}$ $c = 14$

The center $O_1$ is $(-g, -f) = (\sqrt{2}, 3\sqrt{2})$. The radius $R_1$ is $\sqrt{g^2 + f^2 - c} = \sqrt{(-\sqrt{2})^2 + (-3\sqrt{2})^2 - 14} = \sqrt{2 + 18 - 14} = \sqrt{6}$.

The diameter of $C_1$ is $2R_1 = 2\sqrt{6}$.

Step 2: Analyze the second circle ($C_2$)

The equation of the second circle is $(x - 2\sqrt{2})^2 + (y - 2\sqrt{2})^2 = r^2$. This is in the standard form $(x-h)^2 + (y-k)^2 = R^2$. The center $O_2$ is $(2\sqrt{2}, 2\sqrt{2})$. The radius is $R_2 = r$, and we want to find $r^2$.

Step 3: Relate the two circles

A diameter of $C_1$ is a chord of $C_2$. The length of this chord is $L = 2\sqrt{6}$. The midpoint of this chord is the center of $C_1$, which is $O_1 = (\sqrt{2}, 3\sqrt{2})$. The perpendicular distance $d$ from the center of $C_2$ ($O_2$) to the chord is the distance between $O_1$ and $O_2$.

Using the distance formula: $d = \sqrt{(2\sqrt{2} - \sqrt{2})^2 + (2\sqrt{2} - 3\sqrt{2})^2} = \sqrt{(\sqrt{2})^2 + (-\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2$.

Step 4: Apply the chord-radius-distance formula to circle $C_2$

We have $R_2 = r$, $L = 2\sqrt{6}$, and $d = 2$. Using the formula $R_2^2 = d^2 + \left(\frac{L}{2}\right)^2$: $r^2 = 2^2 + \left(\frac{2\sqrt{6}}{2}\right)^2 = 4 + (\sqrt{6})^2 = 4 + 6 = 10$.

Common Mistakes & Tips

• Carefully distinguish between the radii of the two circles.
• Remember that the length of the chord is the diameter of the first circle, which is twice its radius.
• The perpendicular distance is from the center of the larger circle (the one containing the chord) to the chord.

Summary

We found the center and radius of the first circle, then used the fact that a diameter of the first circle is a chord of the second. We calculated the distance between the centers of the two circles, which is the perpendicular distance from the center of the second circle to the chord. Finally, we used the relationship between the radius, chord length, and perpendicular distance to find $r^2 = 10$.

The final answer is $\boxed{10}$.