Find the equation of the hyperbola whose
(i) foci are `(pm 3,0) and ` vertices `(pm 2,0)`
(ii) foci are `(0, pm 8) and ` vertices `(0, pm 5).`

Answer

To find the equations of the hyperbolas given the foci and vertices, we will follow these steps for each case.

### Part (i)
1. **Identify the foci and vertices:**
- Foci: \( (\pm 3, 0) \)
- Vertices: \( (\pm 2, 0) \)

2. **Determine the orientation of the hyperbola:**
- Since the vertices and foci are on the x-axis, the hyperbola opens horizontally. The standard form of the equation for a horizontally opening hyperbola is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

3. **Find \( a \):**
- The distance from the center to the vertices is \( a \). From the vertices \( (\pm 2, 0) \), we have:
\[ a = 2 \]
- Therefore, \( a^2 = 2^2 = 4 \).

4. **Find \( c \):**
- The distance from the center to the foci is \( c \). From the foci \( (\pm 3, 0) \), we have:
\[ c = 3 \]
- Therefore, \( c^2 = 3^2 = 9 \).

5. **Use the relationship between \( a \), \( b \), and \( c \):**
- The relationship is given by:
\[ c^2 = a^2 + b^2 \]
- Substituting the known values:
\[ 9 = 4 + b^2 \]
- Solving for \( b^2 \):
\[ b^2 = 9 - 4 = 5 \]

6. **Write the equation of the hyperbola:**
- Substituting \( a^2 \) and \( b^2 \) into the standard form:
\[ \frac{x^2}{4} - \frac{y^2}{5} = 1 \]

### Part (ii)
1. **Identify the foci and vertices:**
- Foci: \( (0, \pm 8) \)
- Vertices: \( (0, \pm 5) \)

2. **Determine the orientation of the hyperbola:**
- Since the vertices and foci are on the y-axis, the hyperbola opens vertically. The standard form of the equation for a vertically opening hyperbola is:
\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]

3. **Find \( a \):**
- The distance from the center to the vertices is \( a \). From the vertices \( (0, \pm 5) \), we have:
\[ a = 5 \]
- Therefore, \( a^2 = 5^2 = 25 \).

4. **Find \( c \):**
- The distance from the center to the foci is \( c \). From the foci \( (0, \pm 8) \), we have:
\[ c = 8 \]
- Therefore, \( c^2 = 8^2 = 64 \).

5. **Use the relationship between \( a \), \( b \), and \( c \):**
- The relationship is given by:
\[ c^2 = a^2 + b^2 \]
- Substituting the known values:
\[ 64 = 25 + b^2 \]
- Solving for \( b^2 \):
\[ b^2 = 64 - 25 = 39 \]

6. **Write the equation of the hyperbola:**
- Substituting \( a^2 \) and \( b^2 \) into the standard form:
\[ \frac{y^2}{25} - \frac{x^2}{39} = 1 \]

### Final Answers:
- For part (i):
\[ \frac{x^2}{4} - \frac{y^2}{5} = 1 \]
- For part (ii):
\[ \frac{y^2}{25} - \frac{x^2}{39} = 1 \]