Show that the equation `9x^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 solve the problem step by step, we will first rewrite the equation of the hyperbola and then find the required parameters.

### Step 1: Rewrite the equation
The given equation is:
\[ 9x^2 - 16y^2 - 18x + 32y - 151 = 0 \]

### Step 2: Rearrange the equation
Group the \(x\) and \(y\) terms:
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