Which of the following is INCORRECT for the hyperbola `x^(2) - 2y^(2) - 2x +8y-1 = 0`

Options

A. Its eccentricity is `sqrt2`
B. Length of the transverse axis is `2sqrt3`
C. Length of the conjugate axis is `2sqrt6`
D. Latus rectum is `4sqrt3`

Answer

To determine which of the options is incorrect for the hyperbola given by the equation \(x^{2} - 2y^{2} - 2x + 8y - 1 = 0\), we will first simplify the equation and convert it into the standard form of a hyperbola.

### Step-by-Step Solution:

**Step 1: Rearranging the equation**
Start with the original equation:
\[ x^{2} - 2y^{2} - 2x + 8y - 1 = 0 \]
Rearranging gives:
\[ x^{2} - 2x - 2y^{2} + 8y - 1 = 0 \]

**Step 2: Completing the square for \(x\)**
For the \(x\) terms \(x^{2} - 2x\):
\[ x^{2} - 2x = (x - 1)^{2} - 1 \]
So, substituting back, we have:
\[ (x - 1)^{2} - 1 - 2y^{2} + 8y - 1 = 0 \]
This simplifies to:
\[ (x - 1)^{2} - 2y^{2} + 8y - 2 = 0 \]

**Step 3: Completing the square for \(y\)**
Now for the \(y\) terms \(-2y^{2} + 8y\):
Factor out \(-2\):
\[ -2(y^{2} - 4y) = -2((y - 2)^{2} - 4) = -2(y - 2)^{2} + 8 \]
Substituting this back gives:
\[ (x - 1)^{2} - 2(y - 2)^{2} + 8 - 2 = 0 \]
This simplifies to:
\[ (x - 1)^{2} - 2(y - 2)^{2} + 6 = 0 \]
Rearranging gives:
\[ (x - 1)^{2} - 2(y - 2)^{2} = -6 \]

**Step 4: Dividing by -6**
To put it in standard form, divide the entire equation by -6:
\[ -\frac{(x - 1)^{2}}{6} + \frac{(y - 2)^{2}}{3} = 1 \]
This can be rewritten as:
\[ \frac{(y - 2)^{2}}{3} - \frac{(x - 1)^{2}}{6} = 1 \]

**Step 5: Identifying parameters**
From the standard form \(\frac{(y - k)^{2}}{a^{2}} - \frac{(x - h)^{2}}{b^{2}} = 1\), we identify:
- \(a^{2} = 3\) (thus \(a = \sqrt{3}\))
- \(b^{2} = 6\) (thus \(b = \sqrt{6}\))

**Step 6: Finding the value of \(c\)**
Using the relationship \(c^{2} = a^{2} + b^{2}\):
\[ c^{2} = 3 + 6 = 9 \implies c = 3 \]

**Step 7: Calculating eccentricity**
The eccentricity \(e\) of the hyperbola is given by:
\[ e = \frac{c}{a} = \frac{3}{\sqrt{3}} = \sqrt{3} \]

### Conclusion
The eccentricity of the hyperbola is \(\sqrt{3}\). If one of the options states that the eccentricity is \(\sqrt{2}\), then that option is incorrect.