Find the equation to the ellipse with axes as the axes of coordinates.
major axis `9/2` and eccentricity `1/sqrt(3)`, where the major axis is the horizontal axis,

Answer

To find the equation of the ellipse with the given parameters, we will follow these steps:

### Step 1: Identify the given parameters
We know:
- Length of the major axis = \( \frac{9}{2} \)
- Eccentricity \( e = \frac{1}{\sqrt{3}} \)
- The major axis is horizontal, meaning the equation will be of the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

### Step 2: Calculate the semi-major axis \( a \)
The length of the major axis is \( 2a \). Therefore, we can find \( a \) as follows:
\[ 2a = \frac{9}{2} \implies a = \frac{9}{4} \]

### Step 3: Use the relationship between \( a \), \( b \), and \( e \)
We know the relationship:
\[ b^2 = a^2(1 - e^2) \]
First, we need to calculate \( e^2 \):
\[ e^2 = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]

### Step 4: Calculate \( a^2 \)
Next, calculate \( a^2 \):
\[ a^2 = \left(\frac{9}{4}\right)^2 = \frac{81}{16} \]

### Step 5: Substitute \( a^2 \) and \( e^2 \) into the equation for \( b^2 \)
Now substitute \( a^2 \) and \( e^2 \) into the equation for \( b^2 \):
\[ b^2 = \frac{81}{16} \left(1 - \frac{1}{3}\right) = \frac{81}{16} \left(\frac{2}{3}\right) = \frac{81 \times 2}{16 \times 3} = \frac{162}{48} = \frac{27}{8} \]

### Step 6: Write the standard equation of the ellipse
Now we have \( a^2 = \frac{81}{16} \) and \( b^2 = \frac{27}{8} \). The standard form of the ellipse is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Substituting the values:
\[ \frac{x^2}{\frac{81}{16}} + \frac{y^2}{\frac{27}{8}} = 1 \]

### Step 7: Simplify the equation
To simplify, we can multiply through by the least common multiple of the denominators, which is \( 16 \times 8 = 128 \):
\[ \frac{16x^2}{81} + \frac{8y^2}{27} = 1 \]
Multiplying through by \( 128 \):
\[ \frac{128 \cdot 16x^2}{81} + \frac{128 \cdot 8y^2}{27} = 128 \]
This simplifies to:
\[ \frac{2048x^2}{81} + \frac{1024y^2}{27} = 128 \]

### Final Equation
Thus, the equation of the ellipse is:
\[ \frac{x^2}{\frac{81}{16}} + \frac{y^2}{\frac{27}{8}} = 1 \]