Show that the equation `x^2-2y^2-2x+8y-1=0` represents a hyperbola. Find the coordinates of the centre, lengths of the axes, eccentricity, latusrectum, .

Explanation

To show that the equation \( x^2 - 2y^2 - 2x + 8y - 1 = 0 \) represents a hyperbola, we will first rearrange and complete the square for both \( x \) and \( y \).

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