Key Concepts and Formulas

• Standard Equation of a Circle: $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Condition for Touching Circles: Two circles with centers $C_1$ and $C_2$ and radii $r_1$ and $r_2$ touch each other if the distance between their centers is equal to the sum or the absolute difference of their radii: $C_1C_2 = |r_1 \pm r_2|$.

Step-by-Step Solution

Step 1: Find the center and radius of the first circle

The equation of the first circle is $x^2 - 6x + y^2 + 8 = 0$. To find the center and radius, we complete the square to rewrite the equation in standard form. $x^2 - 6x + y^2 + 8 = 0$ $(x^2 - 6x + 9) + y^2 = 9 - 8$ $(x - 3)^2 + (y - 0)^2 = 1$ Comparing this to the standard form $(x-h)^2 + (y-k)^2 = r^2$, we get: Center $C_1 = (3, 0)$ and radius $r_1 = \sqrt{1} = 1$.

Step 2: Find the center and radius of the second circle

The equation of the second circle is $x^2 - 8y + y^2 + 16 - k = 0$. We complete the square to rewrite the equation in standard form. $x^2 + y^2 - 8y + 16 - k = 0$ $x^2 + (y^2 - 8y + 16) = k$ $(x - 0)^2 + (y - 4)^2 = k$ Comparing this to the standard form $(x-h)^2 + (y-k)^2 = r^2$, we get: Center $C_2 = (0, 4)$ and radius $r_2 = \sqrt{k}$.

Step 3: Calculate the distance between the centers of the two circles

Using the distance formula, we find the distance between $C_1(3, 0)$ and $C_2(0, 4)$: $C_1C_2 = \sqrt{(0 - 3)^2 + (4 - 0)^2}$ $C_1C_2 = \sqrt{(-3)^2 + (4)^2}$ $C_1C_2 = \sqrt{9 + 16}$ $C_1C_2 = \sqrt{25}$ $C_1C_2 = 5$

Step 4: Apply the condition for the circles to touch

The circles touch each other, so $C_1C_2 = |r_1 \pm r_2|$. Substituting the values we have: $5 = |1 \pm \sqrt{k}|$ This gives us two possible cases:

Case A: External Touching $5 = 1 + \sqrt{k}$ $\sqrt{k} = 4$ $k = 16$

Case B: Internal Touching $5 = |1 - \sqrt{k}|$ This means either $1 - \sqrt{k} = 5$ or $1 - \sqrt{k} = -5$.

• Subcase B1: $1 - \sqrt{k} = 5$ $-\sqrt{k} = 4$ $\sqrt{k} = -4$ Since $\sqrt{k}$ cannot be negative, this case is invalid.

• Subcase B2: $1 - \sqrt{k} = -5$ $-\sqrt{k} = -6$ $\sqrt{k} = 6$ $k = 36$

Step 5: Determine the largest value of k

The possible values for $k$ are $16$ and $36$. Since we want the largest value of $k$, we choose $k = 36$.

Common Mistakes & Tips

• Sign Errors: Be careful with signs when completing the square and using the distance formula.
• Absolute Value: Remember to consider both positive and negative cases when dealing with the absolute value equation $|1 - \sqrt{k}| = 5$.
• Invalid Solutions: Always check if your solutions are valid. In this case, we had to discard $\sqrt{k} = -4$ because the square root of a real number cannot be negative.

Summary

We found the centers and radii of the two circles, calculated the distance between their centers, and then applied the condition for touching circles. This led to two possible values for $k$, $16$ and $36$. The largest value of $k$ is $36$.

The final answer is $\boxed{36}$.