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 every term 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:
- \(a^2 = 4\) (thus \(a = 2\))
- \(b^2 = 12\) (thus \(b = 2\sqrt{3}\))

### Step 3: Use the formula for the length of the latus rectum
The length of the latus rectum \(L\) of an ellipse is given by the formula:
\[ L = \frac{2b^2}{a} \]
Substituting the values of \(b^2\) and \(a\):
\[ L = \frac{2 \cdot 12}{2} = \frac{24}{2} = 12 \]

### Step 4: Conclusion
Thus, the length of the latus rectum of the ellipse is:
\[ \boxed{12} \]

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