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To solve this problem, we need to find the distance from the center to one of the foci of the hyperbola given by the equation:

\[
\frac{(x - 2)^2}{20} - \frac{(y + 3)^2}{5} = 1
\]

This is the equation of a hyperbola in standard form:

\[
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
\]

Where:
- \((h, k)\) is the center of the hyperbola.
- \(a^2 = 20\), so \(a = \sqrt{20}\).
- \(b^2 = 5\), so \(b = \sqrt{5}\).

For hyperbolas of this form (with a horizontal transverse axis), the distance from the center to a focus (denoted by \(c\)) is given by:

\[
c = \sqrt{a^2 + b^2}
\]

Substituting the values of \(a^2\) and \(b^2\):

\[
c = \sqrt{20 + 5} = \sqrt{25} = 5
\]

So, the correct distance from the center to one of the foci is \(c = 5\) units.

Would you like a detailed explanation or have any further questions?

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Here are 5 related questions for deeper understanding:
1. How do you derive the standard form of a hyperbola's equation?
2. What is the significance of \(a\), \(b\), and \(c\) in the context of hyperbolas?
3. How do you distinguish between vertical and horizontal hyperbolas?
4. What changes in the equation if the hyperbola opens vertically instead of horizontally?
5. How do you find the asymptotes of a hyperbola?

**Tip:** Always check whether the hyperbola opens horizontally or vertically by examining which variable appears first in the standard form equation!

What is the distance from the center to one of the focus of the hyperbola given by the equation: (x−2)^2 / 20 − (y+3)^2 / 5 = 1?

Learn how to find the distance from the center to one of the foci for the hyperbola given by the equation (x−2)^2 / 20 − (y+3)^2 / 5 = 1. This detailed solution covers key concepts like the standard form of a hyperbola and the formula for calculating the distance from the center to a focus. Suitable for high school students studying analytic geometry.

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