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 equation of the ellipse into standard form, we start with the general form of the ellipse provided:

\[
9x^2 + 81y^2 - 729 = 0
\]

### 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 it
\[
\frac{9x^2}{729} + \frac{81y^2}{729} = 1
\]

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

### Step 4: Write the equation in standard form
The standard form of an ellipse is given by:
\[
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
\]

Here, \( h = 0 \), \( k = 0 \), \( a^2 = 81 \), and \( b^2 = 9 \).

So, the equation in standard form is:
\[
\frac{x^2}{9^2} + \frac{y^2}{3^2} = 1
\]
or
\[
\frac{x^2}{81} + \frac{y^2}{9} = 1
\]

This is the standard form of the given ellipse equation.

Would you like further details or have any questions?

### Related Questions:
1. How do you find the center of an ellipse given its equation in standard form?
2. What is the relationship between the lengths of the major and minor axes of an ellipse?
3. How can you determine if an ellipse is vertical or horizontal from its standard form equation?
4. What is the significance of the foci in the ellipse equation, and how are they calculated?
5. How do you convert the standard form of an ellipse into parametric form?

### Tip:
When converting an equation of an ellipse into standard form, always check if the coefficients of \(x^2\) and \(y^2\) are perfect squares. This simplifies the process of identifying the lengths of the major and minor axes.

Learn how to convert the equation of an ellipse, 9x^2 + 81y^2 - 729 = 0, into standard form with a detailed step-by-step solution. This problem covers key geometric concepts related to ellipses, including the standard form formula and simplifying equations. Suitable for high school students studying geometry.

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