Find the equation of the ellipse whose foci are (-1, 5) and (5, 5) and whose major axis is 10.

Answer

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

### Step 1: Identify the foci and the length of the major axis
The foci of the ellipse are given as (-1, 5) and (5, 5). The length of the major axis is given as 10.

### Step 2: Calculate the distance between the foci
The distance between the foci can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of the foci:
\[ d = \sqrt{(5 - (-1))^2 + (5 - 5)^2} = \sqrt{(5 + 1)^2 + 0} = \sqrt{6^2} = 6 \]
This distance \(d\) is equal to \(2c\), where \(c\) is the distance from the center to each focus.

### Step 3: Find the value of \(c\)
Since \(d = 2c\):
\[ 2c = 6 \implies c = 3 \]

### Step 4: Find the value of \(a\)
The length of the major axis is given as 10, which means:
\[ 2a = 10 \implies a = 5 \]

### Step 5: Calculate the value of \(b\)
Using the relationship \(c^2 = a^2 - b^2\):
\[ c^2 = a^2 - b^2 \implies 3^2 = 5^2 - b^2 \implies 9 = 25 - b^2 \]
Rearranging gives:
\[ b^2 = 25 - 9 = 16 \implies b = 4 \]

### Step 6: Find the center of the ellipse
The center of the ellipse is the midpoint of the foci:
\[ \text{Center} = \left(\frac{-1 + 5}{2}, \frac{5 + 5}{2}\right) = \left(\frac{4}{2}, \frac{10}{2}\right) = (2, 5) \]

### Step 7: Write the equation of the ellipse
The standard form of the equation of an ellipse centered at \((h, k)\) is:
\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]
Substituting \(h = 2\), \(k = 5\), \(a^2 = 25\), and \(b^2 = 16\):
\[ \frac{(x-2)^2}{25} + \frac{(y-5)^2}{16} = 1 \]

### Final Answer
The equation of the ellipse is:
\[ \frac{(x-2)^2}{25} + \frac{(y-5)^2}{16} = 1 \]

---