if vertices of an ellipse are `(-4,1),(6,1)` and `x-2y=2` is focal chord then the eccentricity of the ellipse is

Options

A. `(3)/(5)`
B. `(4)/(5)`
C. `(2)/(5)`
D. `(1)/(5)`

Explanation

To find the eccentricity of the ellipse with given vertices and a focal chord, we will follow these steps:

### Step 1: Identify the vertices of the ellipse
The vertices of the ellipse are given as \((-4, 1)\) and \((6, 1)\).

### Step 2: Calculate the length of the major axis
The length of the major axis is the distance between the vertices. We can calculate this as:
\[ \text{Length of major axis} = \text{Distance} = |x_2 - x_1| = |6 - (-4)| = |6 + 4| = 10 \]
Since the length of the major axis is \(2a\), we have:
\[ 2a = 10 \implies a = 5 \]

### Step 3: Find the center of the ellipse
The center of the ellipse can be found by calculating the midpoint of the vertices:
\[ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{-4 + 6}{2}, \frac{1 + 1}{2}\right) = \left(\frac{2}{2}, \frac{2}{2}\right) = (1, 1) \]

### Step 4: Determine the coordinates of the foci
Since the ellipse is horizontal (as the y-coordinates of the vertices are the same), the foci will be located at \((1 - c, 1)\) and \((1 + c, 1)\), where \(c = \sqrt{a^2 - b^2}\).

### Step 5: Use the focal chord information
We are given that the line \(x - 2y = 2\) is a focal chord. We can find the coordinates of one of the foci using the equation of the line.

Substituting \(y = 1\) into the line equation:
\[ x - 2(1) = 2 \implies x - 2 = 2 \implies x = 4 \]
Thus, one of the foci is at \((4, 1)\).

### Step 6: Calculate \(c\)
Now, we know that one focus is at \((4, 1)\) and the center is at \((1, 1)\). The distance \(c\) can be calculated as:
\[ c = |4 - 1| = 3 \]

### Step 7: Use the relationship \(c^2 = a^2 - b^2\) to find \(b\)
We know \(a = 5\) and \(c = 3\):
\[ c^2 = a^2 - b^2 \implies 3^2 = 5^2 - b^2 \implies 9 = 25 - b^2 \implies b^2 = 25 - 9 = 16 \implies b = 4 \]

### Step 8: Calculate the eccentricity \(e\)
The eccentricity \(e\) of the ellipse is given by:
\[ e = \frac{c}{a} = \frac{3}{5} \]

### Final Answer
Thus, the eccentricity of the ellipse is:
\[ \boxed{\frac{3}{5}} \]