Foci are (0, `pm`3), e = `3/4`, equation of the ellipse is

Options

A. `(x^(2))/(7)+(y^(2))/(16)=1`
B. `(x^(2))/(9)+(y^(2))/(16)=1`
C. `(x^(2))/(9)+(y^(2))/(18)=1`
D. `(x^(2))/(16)+(y^(2))/(25)=1`

Answer

To find the equation of the ellipse given the foci and eccentricity, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given information**:
- Foci: (0, ±3)
- Eccentricity (e): 3/4

2. **Determine the orientation of the ellipse**:
- Since the foci are on the y-axis, the major axis of the ellipse is vertical. The standard form of the equation of the ellipse is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
- Here, \( b > a \) because the major axis is vertical.

3. **Use the distance of the foci to find b**:
- The distance of the foci from the center (0,0) is given as 3. Therefore, we have:
\[ c = 3 \]
- For an ellipse, the relationship between a, b, and c is given by:
\[ c^2 = b^2 - a^2 \]

4. **Relate eccentricity to a and b**:
- The eccentricity is defined as:
\[ e = \frac{c}{b} \]
- Substituting the values we have:
\[ \frac{3}{b} = \frac{3}{4} \]
- Cross-multiplying gives:
\[ 3 \cdot 4 = 3b \implies b = 4 \]

5. **Substitute b into the equation for c**:
- Now we have \( b = 4 \) and \( c = 3 \). We can substitute \( b \) into the equation:
\[ c^2 = b^2 - a^2 \]
- This gives:
\[ 3^2 = 4^2 - a^2 \implies 9 = 16 - a^2 \]
- Rearranging gives:
\[ a^2 = 16 - 9 = 7 \]

6. **Write the equation of the ellipse**:
- Now that we have \( a^2 = 7 \) and \( b^2 = 16 \), we can write the equation of the ellipse:
\[ \frac{x^2}{7} + \frac{y^2}{16} = 1 \]

### Final Answer:
The equation of the ellipse is:
\[ \frac{x^2}{7} + \frac{y^2}{16} = 1 \]