Find the equation of the hyperbola satisfying the given conditions :
(i) Vertices `(0, pm 3),` , foci `(0, pm 5) `
(ii) Vertices `(0, pm 5) `, foci `(0 , pm 8 ) `

Answer

To find the equations of the hyperbolas based on the given conditions, we will follow these steps:

### Step 1: Identify the Structure of the Hyperbola
Since the vertices and foci are given on the y-axis, the equation of the hyperbola will take the form:
\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]

### Step 2: First Hyperbola (Vertices at (0, ±3) and Foci at (0, ±5))
1. **Vertices**: The vertices are at (0, ±3), which gives us \( a = 3 \).
- Therefore, \( a^2 = 3^2 = 9 \).

2. **Foci**: The foci are at (0, ±5), which gives us \( c = 5 \).
- Therefore, \( c^2 = 5^2 = 25 \).

3. **Relationship**: We know that for hyperbolas, the relationship between \( a \), \( b \), and \( c \) is given by:
\[ c^2 = a^2 + b^2 \]
Substituting the values we have:
\[ 25 = 9 + b^2 \]
Solving for \( b^2 \):
\[ b^2 = 25 - 9 = 16 \]

4. **Equation of the Hyperbola**: Now substituting \( a^2 \) and \( b^2 \) into the hyperbola equation:
\[ \frac{y^2}{9} - \frac{x^2}{16} = 1 \]

### Step 3: Second Hyperbola (Vertices at (0, ±5) and Foci at (0, ±8))
1. **Vertices**: The vertices are at (0, ±5), which gives us \( a = 5 \).
- Therefore, \( a^2 = 5^2 = 25 \).

2. **Foci**: The foci are at (0, ±8), which gives us \( c = 8 \).
- Therefore, \( c^2 = 8^2 = 64 \).

3. **Relationship**: Using the same relationship:
\[ c^2 = a^2 + b^2 \]
Substituting the values we have:
\[ 64 = 25 + b^2 \]
Solving for \( b^2 \):
\[ b^2 = 64 - 25 = 39 \]

4. **Equation of the Hyperbola**: Now substituting \( a^2 \) and \( b^2 \) into the hyperbola equation:
\[ \frac{y^2}{25} - \frac{x^2}{39} = 1 \]

### Final Results
1. The equation of the first hyperbola is:
\[ \frac{y^2}{9} - \frac{x^2}{16} = 1 \]

2. The equation of the second hyperbola is:
\[ \frac{y^2}{25} - \frac{x^2}{39} = 1 \]