Find the equation of the hyperbola, referred to its axes as the axes of coordinates,
Whose transverse and conjugate axes in length respectively 2 and 3,

Answer

To find the equation of the hyperbola whose transverse and conjugate axes are of lengths 2 and 3 respectively, we can follow these steps:

### Step 1: Identify the lengths of the axes
The lengths of the transverse and conjugate axes are given as:
- Transverse axis = 2
- Conjugate axis = 3

### Step 2: Relate the lengths to 'a' and 'b'
The lengths of the axes are related to 'a' and 'b' as follows:
- The length of the transverse axis is \(2a\)
- The length of the conjugate axis is \(2b\)

From the given lengths:
- \(2a = 2\) implies \(a = 1\)
- \(2b = 3\) implies \(b = \frac{3}{2}\)

### Step 3: Write the standard form of the hyperbola
The standard form of the hyperbola centered at the origin with the transverse axis along the x-axis is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

### Step 4: Substitute the values of 'a' and 'b'
Now substituting \(a = 1\) and \(b = \frac{3}{2}\) into the standard form:
- \(a^2 = 1^2 = 1\)
- \(b^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}\)

Thus, the equation becomes:
\[ \frac{x^2}{1} - \frac{y^2}{\frac{9}{4}} = 1 \]

### Step 5: Simplify the equation
This can be rewritten as:
\[ x^2 - \frac{4y^2}{9} = 1 \]

### Step 6: Clear the fraction
To eliminate the fraction, multiply through by 9:
\[ 9x^2 - 4y^2 = 9 \]

### Final Equation
Thus, the equation of the hyperbola is:
\[ 9x^2 - 4y^2 = 9 \]