Find the eccentricity, the semi-major axis, the semi-minor axis, the coordinates of the foci, the equations of the directrices and the length of the latus rectum of the ellipse `3x^(2)+4y^(2)=12`.

Answer

To solve the problem step by step, we will start with the given equation of the ellipse and convert it into standard form. Then, we will find the required parameters.

### Step 1: Convert the equation into standard form
The given equation is:
\[ 3x^2 + 4y^2 = 12 \]

To convert it into standard form, we divide the entire equation by 12:
\[ \frac{3x^2}{12} + \frac{4y^2}{12} = 1 \]
This simplifies to:
\[ \frac{x^2}{4} + \frac{y^2}{3} = 1 \]

### Step 2: Identify \(a^2\) and \(b^2\)
From the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:
- \(a^2 = 4\) → \(a = \sqrt{4} = 2\)
- \(b^2 = 3\) → \(b = \sqrt{3}\)

### Step 3: Calculate the eccentricity \(e\)
The eccentricity \(e\) is given by the formula:
\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]
Substituting the values of \(a^2\) and \(b^2\):
\[ e = \sqrt{1 - \frac{3}{4}} = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

### Step 4: Find the coordinates of the foci
The coordinates of the foci for an ellipse are given by \((\pm ae, 0)\):
\[ ae = 2 \cdot \frac{1}{2} = 1 \]
Thus, the coordinates of the foci are:
\[ (1, 0) \text{ and } (-1, 0) \]

### Step 5: Find the equations of the directrices
The equations of the directrices are given by:
\[ x = \pm \frac{a}{e} \]
Substituting the values of \(a\) and \(e\):
\[ x = \pm \frac{2}{\frac{1}{2}} = \pm 4 \]
Thus, the equations of the directrices are:
\[ x = 4 \text{ and } x = -4 \]

### Step 6: Calculate the length of the latus rectum
The length of the latus rectum \(L\) is given by the formula:
\[ L = \frac{2b^2}{a} \]
Substituting the values of \(b^2\) and \(a\):
\[ L = \frac{2 \cdot 3}{2} = 3 \]

### Summary of Results
- Eccentricity \(e = \frac{1}{2}\)
- Semi-major axis \(a = 2\)
- Semi-minor axis \(b = \sqrt{3}\)
- Coordinates of the foci: \((1, 0)\) and \((-1, 0)\)
- Equations of the directrices: \(x = 4\) and \(x = -4\)
- Length of the latus rectum: \(3\)