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We are tasked with finding the equation of an ellipse given the following information:

- Center: \( (1, 2) \)
- Horizontal major axis
- Focal distance: 20 units
- Minor axis length: 42 units

### Step 1: General Equation of an Ellipse
The standard equation of an ellipse with a horizontal major axis and center at \( (h, k) \) is:
\[
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
\]
Where:
- \( a \) is the semi-major axis (half of the major axis length).
- \( b \) is the semi-minor axis (half of the minor axis length).
- The foci are located at \( (h \pm c, k) \), where \( c \) is the focal distance from the center.

### Step 2: Calculating \( a \), \( b \), and \( c \)
- The focal distance \( c = 10 \) units (since focal distance is given as 20, and we divide by 2).
- The length of the minor axis is 42 units, so \( b = \frac{42}{2} = 21 \) units.

Using the relationship \( c^2 = a^2 - b^2 \), we can solve for \( a \):
\[
c^2 = a^2 - b^2
\]
\[
10^2 = a^2 - 21^2
\]
\[
100 = a^2 - 441
\]
\[
a^2 = 541
\]
Thus, \( a = \sqrt{541} \).

### Step 3: Equation of the Ellipse
Substitute the values of \( a^2 = 541 \) and \( b^2 = 441 \) into the standard equation:
\[
\frac{(x - 1)^2}{541} + \frac{(y - 2)^2}{441} = 1
\]

This is the equation of the ellipse.

Would you like any further details or clarifications?

Here are five related questions:
1. How do you find the focal points of an ellipse?
2. What is the relationship between the semi-major and semi-minor axes in an ellipse?
3. How do you determine the eccentricity of an ellipse?
4. Can an ellipse have a vertical major axis instead of a horizontal one?
5. How does the equation change for ellipses centered at the origin?

**Tip:** The distance between the foci of an ellipse helps determine how "stretched" it is. This measure is directly linked to the ellipse's eccentricity.

Find the equation of an ellipse with center at (1, 2), horizontal major axis, focal distance of length 20 units, and minor axis of length 42 units.

Find the equation of an ellipse with center at (1, 2), horizontal major axis, focal distance of 20 units, and minor axis of 42 units. This problem involves key concepts from conic sections and geometry. Suitable for high school students in Grades 10-12.

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