Find the coordinate of the foci, vertices, eccentricity and the length of the latus rectum of the hyperbola
`9y ^(2) - 4x ^(2) = 36`

Answer

To solve the problem, we will follow these steps:

1. **Convert the given equation into standard form.**
2. **Identify the values of \(a\), \(b\), and \(c\).**
3. **Find the coordinates of the foci.**
4. **Find the coordinates of the vertices.**
5. **Calculate the eccentricity.**
6. **Determine the length of the latus rectum.**

### Step 1: Convert the given equation into standard form

The given equation is:
\[ 9y^2 - 4x^2 = 36 \]

To convert it into standard form, we divide each term by 36:
\[ \frac{9y^2}{36} - \frac{4x^2}{36} = 1 \]
This simplifies to:
\[ \frac{y^2}{4} - \frac{x^2}{9} = 1 \]

### Step 2: Identify the values of \(a\), \(b\), and \(c\)

From the standard form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), we can identify:
- \(a^2 = 4 \Rightarrow a = 2\)
- \(b^2 = 9 \Rightarrow b = 3\)

Now, we calculate \(c\) using the formula:
\[ c^2 = a^2 + b^2 \]
Substituting the values:
\[ c^2 = 4 + 9 = 13 \Rightarrow c = \sqrt{13} \]

### Step 3: Find the coordinates of the foci

For a hyperbola of the form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), the coordinates of the foci are given by:
\[ (0, \pm c) \]
Thus, the coordinates of the foci are:
\[ (0, \sqrt{13}) \quad \text{and} \quad (0, -\sqrt{13}) \]

### Step 4: Find the coordinates of the vertices

The coordinates of the vertices are given by:
\[ (0, \pm a) \]
Thus, the coordinates of the vertices are:
\[ (0, 2) \quad \text{and} \quad (0, -2) \]

### Step 5: Calculate the eccentricity

The eccentricity \(e\) is given by:
\[ e = \frac{c}{a} \]
Substituting the values:
\[ e = \frac{\sqrt{13}}{2} \]

### Step 6: Determine the length of the latus rectum

The length of the latus rectum \(L\) is given by:
\[ L = \frac{2b^2}{a} \]
Substituting the values:
\[ L = \frac{2 \cdot 9}{2} = 9 \]

### Final Results

- **Coordinates of the foci:** \((0, \sqrt{13})\) and \((0, -\sqrt{13})\)
- **Coordinates of the vertices:** \((0, 2)\) and \((0, -2)\)
- **Eccentricity:** \(e = \frac{\sqrt{13}}{2}\)
- **Length of the latus rectum:** \(L = 9\)