The length of the hyperbola of the conjugate axis of `2x^(2) - 3y^(2) =6` is

Options

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

Answer

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

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

### Step 2: Identify \(a^2\) and \(b^2\)
From the standard form of the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), we can identify:
- \(a^2 = 3\)
- \(b^2 = 2\)

### Step 3: Calculate \(b\)
To find \(b\), we take the square root of \(b^2\):
\[ b = \sqrt{2} \]

### Step 4: Find the length of the conjugate axis
The length of the conjugate axis of a hyperbola is given by the formula \(2b\):
\[ \text{Length of conjugate axis} = 2b = 2 \times \sqrt{2} = 2\sqrt{2} \]

### Conclusion
Thus, the length of the conjugate axis of the hyperbola is:
\[ \boxed{2\sqrt{2}} \]

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