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 and directrix, we can follow these steps:

### Step 1: Identify the Focus and Directrix
The focus of the ellipse is given as \( F(-1, -1) \) and the directrix is given by the equation \( x - y + 3 = 0 \).

### Step 2: Rewrite the Directrix Equation
We can rewrite the directrix equation in slope-intercept form:
\[ y = x + 3 \]
This indicates that the slope of the directrix is 1.

### Step 3: Find the Slope of the Axis of the Ellipse
Since the axis of the ellipse is perpendicular to the directrix, the slope of the axis will be the negative reciprocal of the slope of the directrix. Thus, the slope of the axis is:
\[ -\frac{1}{1} = -1 \]

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

### Step 5: Find the Intersection of the Directrix and the Axis
To find the center of the ellipse, we need to find the intersection of the directrix and the axis. We set the equations equal to each other:
\[ x + 3 = -x - 2 \]
Solving for \( x \):
\[ 2x = -5 \implies x = -\frac{5}{2} \]

### Step 6: Substitute \( x \) Back to Find \( y \)
Now substituting \( x = -\frac{5}{2} \) into the equation of the axis to find \( y \):
\[ y = -\left(-\frac{5}{2}\right) - 2 = \frac{5}{2} - 2 = \frac{5}{2} - \frac{4}{2} = \frac{1}{2} \]

### Step 7: Coordinates of the Center
Thus, the coordinates of the center of the ellipse are:
\[ \left(-\frac{5}{2}, \frac{1}{2}\right) \]

### Final Answer
The coordinates of the center of the ellipse are:
\[ \boxed{\left(-\frac{5}{2}, \frac{1}{2}\right)} \]