An ellipse has foci at `F_1(9, 20)` and `F_2(49,55)` in the xy-plane and is tangent to the x-axis. Find the length of its major axis.

Options

A. 85
B. 75
C. 65
D. 55

Explanation

To find the length of the major axis of the ellipse with foci at \( F_1(9, 20) \) and \( F_2(49, 55) \) that is tangent to the x-axis, we can follow these steps:

### Step 1: Determine the coordinates of the foci
The foci of the ellipse are given as:
- \( F_1(9, 20) \)
- \( F_2(49, 55) \)

### Step 2: Calculate the distance between the foci
The distance \( d \) between the two 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:
\[ d = \sqrt{(49 - 9)^2 + (55 - 20)^2} \]
Calculating the differences:
\[ d = \sqrt{(40)^2 + (35)^2} \]
Calculating the squares:
\[ d = \sqrt{1600 + 1225} = \sqrt{2825} \]

### Step 3: Simplify the distance
We can simplify \( \sqrt{2825} \):
\[ \sqrt{2825} = \sqrt{25 \times 113} = 5\sqrt{113} \]

### Step 4: Relate the distance to the major axis
For an ellipse, the distance between the foci \( 2c \) is related to the length of the major axis \( 2a \) by the equation:
\[ 2c = d \]
Thus, we have:
\[ d = 2a \]
Therefore, the length of the major axis \( 2a \) is equal to the distance between the foci:
\[ 2a = 5\sqrt{113} \]

### Step 5: Find the length of the major axis
The length of the major axis \( a \) is:
\[ a = \frac{5\sqrt{113}}{2} \]

### Final Answer
The length of the major axis of the ellipse is:
\[ 5\sqrt{113} \]