If `(0, 3+sqrt5)` is a point on the ellipse whose foci and (2, 3) and `(-2, 3)`, then the length of the semi - major axis is

Answer

To find the length of the semi-major axis of the ellipse given the point \( (0, 3 + \sqrt{5}) \) and the foci \( (2, 3) \) and \( (-2, 3) \), we can follow these steps:

### Step 1: Identify the foci and the point
The foci of the ellipse are given as \( F_1(2, 3) \) and \( F_2(-2, 3) \). The point on the ellipse is \( P(0, 3 + \sqrt{5}) \).

### Step 2: Use the property of ellipses
For any point \( P \) on an ellipse, the sum of the distances from \( P \) to the foci \( F_1 \) and \( F_2 \) is constant and equal to \( 2a \), where \( a \) is the length of the semi-major axis.

### Step 3: Calculate the distances \( PF_1 \) and \( PF_2 \)
Using the distance formula, we can calculate \( PF_1 \) and \( PF_2 \):

1. **Distance \( PF_1 \)**:
\[ PF_1 = \sqrt{(0 - 2)^2 + ((3 + \sqrt{5}) - 3)^2} = \sqrt{(-2)^2 + (\sqrt{5})^2} = \sqrt{4 + 5} = \sqrt{9} = 3 \]

2. **Distance \( PF_2 \)**:
\[ PF_2 = \sqrt{(0 + 2)^2 + ((3 + \sqrt{5}) - 3)^2} = \sqrt{(2)^2 + (\sqrt{5})^2} = \sqrt{4 + 5} = \sqrt{9} = 3 \]

### Step 4: Sum the distances
Now, we sum the distances:
\[ PF_1 + PF_2 = 3 + 3 = 6 \]

### Step 5: Relate the sum to \( 2a \)
Since \( PF_1 + PF_2 = 2a \), we have:
\[ 2a = 6 \]

### Step 6: Solve for \( a \)
To find the semi-major axis \( a \):
\[ a = \frac{6}{2} = 3 \]

### Conclusion
The length of the semi-major axis is \( \boxed{3} \).