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Let's go through the problems step by step.

### Problem 1: Identifying Properties of the Ellipse and Graphing
We are given two equations of ellipses.

#### a. 
\[
\frac{(x-1)^2}{49} + \frac{(y-3)^2}{81} = 1
\]

This is the standard form of an ellipse equation: 
\[
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
\]
where \((h, k)\) is the center, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.

For the given equation:
- Center: \((h, k) = (1, 3)\)
- \(a^2 = 81\), so \(a = 9\)
- \(b^2 = 49\), so \(b = 7\)
- Since \(a > b\), the major axis is vertical.

The ellipse's major axis length is \(2a = 18\) and the minor axis length is \(2b = 14\).

#### b. 
\[
\frac{x^2}{36} + \frac{y^2}{4} = 1
\]

For this equation:
- Center: \((h, k) = (0, 0)\) (since there are no shifts in \(x\) and \(y\))
- \(a^2 = 36\), so \(a = 6\)
- \(b^2 = 4\), so \(b = 2\)
- Since \(a > b\), the major axis is horizontal.

The ellipse's major axis length is \(2a = 12\) and the minor axis length is \(2b = 4\).

### Problem 2: Writing the Standard Form of the Ellipse Equation

We are given:
- Foci \(F_1(0, 0)\) and \(F_2(8, 0)\)
- The major axis length is 16.

The standard form of the ellipse equation is given by:
\[
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
\]
Where:
- The center \((h, k)\) is the midpoint of the foci, so \((h, k) = (4, 0)\).
- The distance between the foci is \(2c = 8\), so \(c = 4\).
- The length of the major axis is \(2a = 16\), so \(a = 8\).
- We find \(b\) using the equation \(c^2 = a^2 - b^2\):
\[
4^2 = 8^2 - b^2 \implies 16 = 64 - b^2 \implies b^2 = 48 \implies b = \sqrt{48} = 4\sqrt{3}
\]

Thus, the standard form of the ellipse is:
\[
\frac{(x-4)^2}{64} + \frac{y^2}{48} = 1
\]

Would you like more details on any of these steps or help with any additional questions?

### Related Questions:
1. How do you determine if the major axis of an ellipse is horizontal or vertical?
2. What is the relationship between the foci, center, and vertices of an ellipse?
3. How does the length of the major axis affect the equation of the ellipse?
4. Can you describe the steps to graph an ellipse given its standard form equation?
5. How do you find the eccentricity of an ellipse?

### Tip:
Remember that the distance from the center to each focus (c) in an ellipse relates to the major and minor axes through the equation \(c^2 = a^2 - b^2\). This is key in understanding the shape and orientation of the ellipse.

1. Identify the properties of the ellipse, then graph.
    a. (x-1)^2/49 + (y-3)^2/81 = 1
    b. x^2/36 + y^2/4 = 1

2. Write the standard form of the equation of ellipse given that F1(0,0) and F2(8,0) and has a major axis of length 16.

Explore how to identify the properties of an ellipse from its equation and write the standard form of an ellipse equation given foci and the length of the major axis. Detailed steps and mathematical concepts are explained, suitable for high school students.

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