Find the length of the axes, the co-ordinates of the foci, the eccentricity, and latus rectum of the ellipse `3x^(2)+2y^(2)=24`.

Answer

To solve the problem step by step, we will analyze the given equation of the ellipse and derive the required values.

### Step 1: Write the equation in standard form
The given equation is:
\[ 3x^2 + 2y^2 = 24 \]

To convert it to standard form, we divide the entire equation by 24:
\[ \frac{3x^2}{24} + \frac{2y^2}{24} = 1 \]

This simplifies to:
\[ \frac{x^2}{8} + \frac{y^2}{12} = 1 \]

### Step 2: Identify \(a^2\) and \(b^2\)
From the standard form of the ellipse:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
we can identify:
- \(a^2 = 8\) (for the x-term)
- \(b^2 = 12\) (for the y-term)

### Step 3: Calculate \(a\) and \(b\)
Now, we take the square roots to find \(a\) and \(b\):
- \(a = \sqrt{8} = 2\sqrt{2}\)
- \(b = \sqrt{12} = 2\sqrt{3}\)

### Step 4: Determine the lengths of the axes
The lengths of the axes of the ellipse are given by:
- Length of the major axis = \(2b = 2 \times 2\sqrt{3} = 4\sqrt{3}\)
- Length of the minor axis = \(2a = 2 \times 2\sqrt{2} = 4\sqrt{2}\)

### Step 5: Calculate the eccentricity \(e\)
The eccentricity of the ellipse is calculated using the formula:
\[ e = \sqrt{1 - \frac{a^2}{b^2}} \]

Substituting the values:
\[ e = \sqrt{1 - \frac{8}{12}} = \sqrt{1 - \frac{2}{3}} = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \]

### Step 6: Find the coordinates of the foci
The coordinates of the foci for an ellipse where \(b > a\) are given by:
\[ (0, \pm be) \]
Calculating \(be\):
\[ be = 2\sqrt{3} \times \frac{1}{\sqrt{3}} = 2 \]

Thus, the coordinates of the foci are:
\[ (0, 2) \text{ and } (0, -2) \]

### Step 7: Calculate the length of the latus rectum
The length of the latus rectum for an ellipse where \(b > a\) is given by:
\[ \frac{2a^2}{b} \]

Calculating:
\[ \frac{2 \times 8}{2\sqrt{3}} = \frac{16}{2\sqrt{3}} = \frac{8}{\sqrt{3}} \]

### Summary of Results
- Length of the major axis: \(4\sqrt{3}\)
- Length of the minor axis: \(4\sqrt{2}\)
- Eccentricity: \(\frac{1}{\sqrt{3}}\)
- Coordinates of the foci: \((0, 2)\) and \((0, -2)\)
- Length of the latus rectum: \(\frac{8}{\sqrt{3}}\)