For the ellipse `3x ^(2) + 4y ^(2) =12, ` the length of latus rectum is

Options

A. `3/2`
B. 3
C. `8/3`
D. `sqrt((3)/(2))`

Answer

To find the length of the latus rectum of the ellipse given by the equation \(3x^2 + 4y^2 = 12\), we can follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the ellipse:
\[ 3x^2 + 4y^2 = 12 \]
To convert this 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:
- \(a^2 = 4\) (thus \(a = 2\))
- \(b^2 = 3\) (thus \(b = \sqrt{3}\))

### Step 3: Determine the length of the latus rectum
For an ellipse, the length of the latus rectum \(L\) is given by the formula:
\[ L = \frac{2b^2}{a} \]
Substituting the values of \(b^2\) and \(a\):
\[ L = \frac{2 \cdot 3}{2} \]
This simplifies to:
\[ L = \frac{6}{2} = 3 \]

### Conclusion
The length of the latus rectum of the ellipse \(3x^2 + 4y^2 = 12\) is \(3\).

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