The length of the latusrectum of the ellipse `3x^(2)+y^(2)=12` is

Options

A. 4
B. 3
C. 8
D. `4/sqrt3`

Explanation

To find the length of the latus rectum of the ellipse given by the equation \(3x^2 + y^2 = 12\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the ellipse:
\[ 3x^2 + y^2 = 12 \]
To convert this into standard form, we divide the entire equation by 12:
\[ \frac{3x^2}{12} + \frac{y^2}{12} = 1 \]
This simplifies to:
\[ \frac{x^2}{4} + \frac{y^2}{12} = 1 \]

### 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:
\[ a^2 = 4 \quad \text{and} \quad b^2 = 12 \]
Thus, we have:
\[ a = \sqrt{4} = 2 \quad \text{and} \quad b = \sqrt{12} = 2\sqrt{3} \]

### Step 3: Determine the length of the latus rectum
For an ellipse where \(b > a\), the length of the latus rectum \(L\) is given by the formula:
\[ L = \frac{2a^2}{b} \]
Substituting the values of \(a^2\) and \(b\):
\[ L = \frac{2 \cdot 4}{2\sqrt{3}} = \frac{8}{2\sqrt{3}} = \frac{4}{\sqrt{3}} \]

### Conclusion
The length of the latus rectum of the ellipse \(3x^2 + y^2 = 12\) is:
\[ \frac{4}{\sqrt{3}} \]

### Final Answer
Thus, the correct option is \(4\) (option 4).

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