The focus of an ellipse is (-1, -1) and the corresponding directrix is `x - y + 3 = 0`. If the eccentricity of the ellipse is 1/2, then the coordinates of the centre of the ellipse, are

Options

A. (1/2, 3/2)
B. (-1/2, 3/2)
C. (-1/2, -3/2)
D. none of these

Explanation

To find the coordinates of the center of the ellipse given the focus, directrix, and eccentricity, we can follow these steps:

### Step 1: Identify the given information
- Focus \( F = (-1, -1) \)
- Directrix: \( x - y + 3 = 0 \)
- Eccentricity \( e = \frac{1}{2} \)

### Step 2: Find the slope of the directrix
The equation of the directrix can be rewritten in slope-intercept form:
\[ y = x + 3 \]
From this, we see that the slope \( m \) of the directrix is \( 1 \).

### Step 3: Find the slope of the axis of the ellipse
The axis of the ellipse is perpendicular to the directrix. Therefore, the slope of the axis \( m_a \) is:
\[ m_a = -\frac{1}{m} = -1 \]

### Step 4: Write the equation of the axis
Using the point-slope form of the line equation, we can write the equation of the axis passing through the focus \( F(-1, -1) \):
\[ y - (-1) = -1(x - (-1)) \]
Simplifying this:
\[ y + 1 = -1(x + 1) \]
\[ y + 1 = -x - 1 \]
\[ x + y + 2 = 0 \]
This is the equation of the axis.

### Step 5: Find the foot of the directrix
To find the foot of the directrix \( D \), we need to solve the system of equations formed by the directrix and the axis:
1. Directrix: \( x - y + 3 = 0 \)
2. Axis: \( x + y + 2 = 0 \)

Adding these two equations:
\[ (x - y + 3) + (x + y + 2) = 0 \]
This simplifies to:
\[ 2x + 5 = 0 \]
Thus:
\[ x = -\frac{5}{2} \]

Now, substituting \( x = -\frac{5}{2} \) back into the equation of the directrix to find \( y \):
\[ -\frac{5}{2} - y + 3 = 0 \]
\[ -y = \frac{5}{2} - 3 \]
\[ -y = \frac{5}{2} - \frac{6}{2} \]
\[ -y = -\frac{1}{2} \]
Thus:
\[ y = \frac{1}{2} \]

So, the coordinates of the foot of the directrix \( D \) are:
\[ D = \left(-\frac{5}{2}, \frac{1}{2}\right) \]

### Step 6: Use the section formula to find the center
Let the center of the ellipse be \( C(x_1, y_1) \). The ratio \( CF: FD = 1:3 \).

Using the section formula, we have:
\[ x_1 = \frac{3(-1) + 1\left(-\frac{5}{2}\right)}{3 + 1} = \frac{-3 - \frac{5}{2}}{4} = \frac{-\frac{6}{2} - \frac{5}{2}}{4} = \frac{-\frac{11}{2}}{4} = -\frac{11}{8} \]

For the y-coordinate:
\[ y_1 = \frac{3(-1) + 1\left(\frac{1}{2}\right)}{3 + 1} = \frac{-3 + \frac{1}{2}}{4} = \frac{-\frac{6}{2} + \frac{1}{2}}{4} = \frac{-\frac{5}{2}}{4} = -\frac{5}{8} \]

### Step 7: Final coordinates of the center
Thus, the coordinates of the center of the ellipse are:
\[ C\left(-\frac{11}{8}, -\frac{5}{8}\right) \]