Let the eccentricity of the hyperbola with the principal axes along the coordinate axes and passing through (3, 0) and `(3sqrt2,2)` is e, then the value of `((e^(2)+1)/(e^(2)-1))` is equal to

Answer

To solve the problem step by step, we will follow the outlined process to find the eccentricity of the hyperbola and then calculate the required expression.

### Step 1: Write the standard equation of the hyperbola
The standard equation of a hyperbola with its principal axes along the coordinate axes is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

### Step 2: Use the first point (3, 0)
Substituting the point (3, 0) into the equation:
\[ \frac{3^2}{a^2} - \frac{0^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{9}{a^2} = 1 \implies a^2 = 9 \implies a = 3 \]

### Step 3: Use the second point (3√2, 2)
Now substituting the second point (3√2, 2) into the equation:
\[ \frac{(3\sqrt{2})^2}{9} - \frac{2^2}{b^2} = 1 \]
Calculating \( (3\sqrt{2})^2 \):
\[ \frac{18}{9} - \frac{4}{b^2} = 1 \implies 2 - \frac{4}{b^2} = 1 \]
Rearranging gives:
\[ 2 - 1 = \frac{4}{b^2} \implies 1 = \frac{4}{b^2} \implies b^2 = 4 \implies b = 2 \]

### Step 4: Calculate the eccentricity \( e \)
The eccentricity \( e \) of a hyperbola is given by the formula:
\[ e = \sqrt{1 + \frac{b^2}{a^2}} \]
Substituting the values of \( a^2 \) and \( b^2 \):
\[ e = \sqrt{1 + \frac{4}{9}} = \sqrt{\frac{9 + 4}{9}} = \sqrt{\frac{13}{9}} = \frac{\sqrt{13}}{3} \]

### Step 5: Calculate \( \frac{e^2 + 1}{e^2 - 1} \)
First, calculate \( e^2 \):
\[ e^2 = \left(\frac{\sqrt{13}}{3}\right)^2 = \frac{13}{9} \]
Now calculate \( e^2 + 1 \) and \( e^2 - 1 \):
\[ e^2 + 1 = \frac{13}{9} + 1 = \frac{13}{9} + \frac{9}{9} = \frac{22}{9} \]
\[ e^2 - 1 = \frac{13}{9} - 1 = \frac{13}{9} - \frac{9}{9} = \frac{4}{9} \]
Now substitute these into the expression:
\[ \frac{e^2 + 1}{e^2 - 1} = \frac{\frac{22}{9}}{\frac{4}{9}} = \frac{22}{4} = \frac{11}{2} = 5.5 \]

### Final Answer
The value of \( \frac{e^2 + 1}{e^2 - 1} \) is:
\[ \boxed{5.5} \]