The length of the latus rectum of `3x^(2) - 2y^(2) =6` is

Options

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

Answer

To find the length of the latus rectum of the hyperbola given by the equation \(3x^2 - 2y^2 = 6\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the hyperbola:
\[ 3x^2 - 2y^2 = 6 \]
To convert this into the standard form of a hyperbola, we divide both sides by 6:
\[ \frac{3x^2}{6} - \frac{2y^2}{6} = 1 \]
This simplifies to:
\[ \frac{x^2}{2} - \frac{y^2}{3} = 1 \]

### 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:
- \(a^2 = 2\)
- \(b^2 = 3\)

### Step 3: Calculate \(a\) and \(b\)
Now, we find \(a\) and \(b\):
\[ a = \sqrt{2}, \quad b = \sqrt{3} \]

### Step 4: Use the formula for the length of the latus rectum
The formula for the length of the latus rectum \(L\) of a hyperbola is given by:
\[ L = \frac{2b^2}{a} \]
Substituting the values we found:
\[ L = \frac{2 \cdot 3}{\sqrt{2}} = \frac{6}{\sqrt{2}} \]

### Step 5: Simplify the expression
To simplify \(\frac{6}{\sqrt{2}}\), we can multiply the numerator and denominator by \(\sqrt{2}\):
\[ L = \frac{6\sqrt{2}}{2} = 3\sqrt{2} \]

### Final Answer
Thus, the length of the latus rectum is:
\[ \boxed{3\sqrt{2}} \]