The length of latus rectum of the ellipse `3x^(2) + y^(2) = 12 ` is (i) 4 (ii) 3 (iii) 8 (iv) `(4)/(sqrt(3))`

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 in standard form
We start with the given equation:
\[ 3x^2 + y^2 = 12 \]
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
The formula for the length of the latus rectum \(L\) of an ellipse is given by:
\[ L = \frac{2a^2}{b} \]
Substituting the values of \(a^2\) and \(b\):
\[ L = \frac{2 \cdot 4}{\sqrt{12}} = \frac{8}{\sqrt{12}} \]
We can simplify \(\sqrt{12}\):
\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \]
Thus, we have:
\[ L = \frac{8}{2\sqrt{3}} = \frac{4}{\sqrt{3}} \]

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

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