You are correct; I made an error in my calculations when combining constants on the right side of the equation. Let's correct that step.

### Correcting the Combination of Constants

When we completed the square for the terms involving \(x\) and \(y\), we had:

\[
(x + 5)^2 - 25 + (y - 4)^2 - 16 = \frac{5}{4}
\]

So, the equation should have been:

\[
(x + 5)^2 + (y - 4)^2 - 41 = \frac{5}{4}
\]

We need to combine the constants correctly:

\[
(x + 5)^2 + (y - 4)^2 = \frac{5}{4} + 41
\]

Now, \(41\) as a fraction with a denominator of 4 is:

\[
41 = \frac{41 \times 4}{4} = \frac{164}{4}
\]

So, adding these fractions:

\[
\frac{5}{4} + \frac{164}{4} = \frac{5 + 164}{4} = \frac{169}{4}
\]

Therefore, the equation becomes:

\[
(x + 5)^2 + (y - 4)^2 = \frac{169}{4}
\]

### Final Result

- The center \((h, k)\) is \((-5, 4)\).
- The radius \(r\) is \(\sqrt{\frac{169}{4}} = \frac{13}{2}\).

So, the previous calculations were correct; the misunderstanding was in the explanation of the steps.

Would you like more details or have any questions about this solution? Here are some related questions you might find helpful:

1. How do you complete the square for a quadratic expression?
2. What is the standard form of a circle's equation?
3. How do you derive the distance formula from the circle equation?
4. What is the geometric interpretation of completing the square?
5. How can you convert a general quadratic equation to the standard form of a circle?
6. What are the properties of a circle in the coordinate plane?
7. How do you determine the equation of a circle given its center and radius?
8. How does the equation of a circle change if it is translated in the coordinate plane?

**Tip:** When completing the square, always remember to add and subtract the same value inside the equation to maintain equality.

Explore how to derive and solve circle equations using completing the square method. Learn step-by-step with an example involving the correction of constants. Suitable for high school students studying algebra and geometry.

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