Find the equation of the ellipse whose minor axis is 4 and which has a distance of 6 units between foci.

Answer

To find the equation of the ellipse whose minor axis is 4 and which has a distance of 6 units between the foci, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Information**:
- The minor axis length is given as 4. Therefore, the semi-minor axis \( b \) can be calculated as:
\[ 2b = 4 \implies b = \frac{4}{2} = 2 \]

2. **Calculate the Distance Between Foci**:
- The distance between the foci is given as 6 units. This means:
\[ 2c = 6 \implies c = \frac{6}{2} = 3 \]

3. **Use the Relationship Between \( a \), \( b \), and \( c \)**:
- For an ellipse, the relationship between the semi-major axis \( a \), semi-minor axis \( b \), and the distance to the foci \( c \) is given by the equation:
\[ c^2 = a^2 - b^2 \]
- We already have \( b = 2 \) and \( c = 3 \). Now we can calculate \( b^2 \) and \( c^2 \):
\[ b^2 = 2^2 = 4 \]
\[ c^2 = 3^2 = 9 \]

4. **Substituting Values to Find \( a^2 \)**:
- Substitute \( b^2 \) and \( c^2 \) into the relationship:
\[ 9 = a^2 - 4 \]
- Rearranging gives:
\[ a^2 = 9 + 4 = 13 \]

5. **Write the Standard Equation of the Ellipse**:
- The standard form of the equation of an ellipse centered at the origin is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
- Substituting \( a^2 = 13 \) and \( b^2 = 4 \):
\[ \frac{x^2}{13} + \frac{y^2}{4} = 1 \]

### Final Answer:
The equation of the ellipse is:
\[ \frac{x^2}{13} + \frac{y^2}{4} = 1 \]