Show that the equation 9`x^2-16 y^2-18 x+32 y-151=0` represents a hyperbola. Find the coordinates of the centre, lengths of the axes, eccentricity, latus-rectum, coordinates of foci and vertices, equations of the directrices of the hyperbola.

Explanation

To show that the equation \( 9x^2 - 16y^2 - 18x + 32y - 151 = 0 \) represents a hyperbola and to find its various characteristics, we can follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ 9x^2 - 16y^2 - 18x + 32y - 151 = 0 \]
Rearranging gives:
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