Answer the questions as instructed.
Find the length of the axes of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` , which passes through the points (3,0) and `(3sqrt(2),2)`.

Answer

To find the lengths of the axes of the hyperbola given by the equation

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

that passes through the points (3, 0) and \((3\sqrt{2}, 2)\), we will follow these steps:

### Step 1: Substitute the first point (3, 0) into the hyperbola equation.

Substituting \(x = 3\) and \(y = 0\) into the hyperbola equation gives:

\[ \frac{3^2}{a^2} - \frac{0^2}{b^2} = 1 \]

This simplifies to:

\[ \frac{9}{a^2} = 1 \]

### Step 2: Solve for \(a^2\).

From the equation \(\frac{9}{a^2} = 1\), we can solve for \(a^2\):

\[ a^2 = 9 \]

Taking the square root gives:

\[ a = 3 \]

### Step 3: Substitute the second point \((3\sqrt{2}, 2)\) into the hyperbola equation.

Now we substitute \(x = 3\sqrt{2}\) and \(y = 2\) into the hyperbola equation:

\[ \frac{(3\sqrt{2})^2}{a^2} - \frac{2^2}{b^2} = 1 \]

### Step 4: Substitute \(a^2\) into the equation.

Since we found \(a^2 = 9\), we substitute this into the equation:

\[ \frac{18}{9} - \frac{4}{b^2} = 1 \]

This simplifies to:

\[ 2 - \frac{4}{b^2} = 1 \]

### Step 5: Solve for \(b^2\).

Rearranging gives:

\[ 2 - 1 = \frac{4}{b^2} \]

\[ 1 = \frac{4}{b^2} \]

Cross-multiplying gives:

\[ b^2 = 4 \]

Taking the square root gives:

\[ b = 2 \]

### Step 6: Find the lengths of the axes.

The lengths of the transverse and conjugate axes are given by \(2a\) and \(2b\) respectively:

\[ \text{Length of transverse axis} = 2a = 2 \times 3 = 6 \]

\[ \text{Length of conjugate axis} = 2b = 2 \times 2 = 4 \]

### Final Answer:

The lengths of the axes of the hyperbola are:

- Transverse axis: 6
- Conjugate axis: 4

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