Solve for the standard form of the living general form of the ellipse nine x ^2 + 80 one y ^2 - 729 = 0.

To convert the given general form of the ellipse equation into the standard form, follow these steps:

### Given Equation:
\[
9x^2 + 81y^2 - 729 = 0
\]

### Step 1: Move the constant term to the right side of the equation.
\[
9x^2 + 81y^2 = 729
\]

### Step 2: Divide the entire equation by 729 to simplify.
\[
\frac{9x^2}{729} + \frac{81y^2}{729} = 1
\]

### Step 3: Simplify each term.
\[
\frac{x^2}{81} + \frac{y^2}{9} = 1
\]

### Step 4: Write the equation in standard form.
The standard form of an ellipse is:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
Here, we have \(a^2 = 81\) and \(b^2 = 9\), so:
\[
\frac{x^2}{9^2} + \frac{y^2}{3^2} = 1
\]

### Final Answer:
The standard form of the ellipse is:
\[
\frac{x^2}{81} + \frac{y^2}{9} = 1
\]

Would you like further details or explanations on any part of this solution? Here are some related questions:

1. How do you find the vertices and foci of the ellipse?
2. What is the major axis of the ellipse?
3. How would you graph this ellipse?
4. How does changing the values of \(a^2\) and \(b^2\) affect the shape of the ellipse?
5. How do you determine if an equation represents an ellipse, a circle, a hyperbola, or a parabola?

**Tip:** The length of the major axis in an ellipse is \(2a\), and the length of the minor axis is \(2b\).

Learn how to convert the general form of the ellipse equation 9x^2 + 81y^2 - 729 = 0 into its standard form. This problem covers conic sections and the standard form of ellipses, suitable for advanced high school students.

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