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

Options

A. 4
B. 3
C. 8
D. `4 //sqrt(3)`

Answer

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

### Step 1: Rewrite the equation of the ellipse in standard form
We start with the equation:
\[ 3x^2 + y^2 = 12 \]
To convert it 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 the values of \(a\) and \(b\)
From the standard form of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify:
- \(a^2 = 4\) (which means \(a = 2\))
- \(b^2 = 12\) (which means \(b = \sqrt{12} = 2\sqrt{3}\))

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

### Step 4: Substitute the values of \(a\) and \(b\) into the formula
Now, we substitute \(a^2\) and \(b\) into the formula:
\[ L = \frac{2 \cdot 4}{2\sqrt{3}} = \frac{8}{2\sqrt{3}} = \frac{4}{\sqrt{3}} \]

### Step 5: Final answer
Thus, the length of the latus rectum of the ellipse \(3x^2 + y^2 = 12\) is:
\[ \frac{4}{\sqrt{3}} \]

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