The circumcentre of the triangle formed by the lines, `xy + 2x + 2y + 4 = 0 and x + y + 2 = 0` is-

Options

A. ( 0,0)
B. ( -2,-2)
C. (-1,-1)
D. (-1,-2)

Answer

To find the circumcenter of the triangle formed by the lines \(xy + 2x + 2y + 4 = 0\) and \(x + y + 2 = 0\), we will follow these steps:

### Step 1: Rewrite the first equation
The first equation is given as:
\[ xy + 2x + 2y + 4 = 0 \]
We can factor this equation. It can be rewritten as:
\[ (x + 2)(y + 2) = 0 \]
This implies that:
\[ x + 2 = 0 \quad \text{or} \quad y + 2 = 0 \]
Thus, we have two lines:
1. \(x + 2 = 0\) (which is the vertical line \(x = -2\))
2. \(y + 2 = 0\) (which is the horizontal line \(y = -2\))

### Step 2: Write the second equation
The second equation is:
\[ x + y + 2 = 0 \]
This can be rearranged to:
\[ y = -x - 2 \]

### Step 3: Find the points of intersection
Now, we will find the intersection points of these lines to form the vertices of the triangle.

1. **Intersection of \(x = -2\) and \(y = -2\)**:
- This gives the point \((-2, -2)\).

2. **Intersection of \(x = -2\) and \(y = -x - 2\)**:
- Substitute \(x = -2\) into \(y = -x - 2\):
\[ y = -(-2) - 2 = 2 - 2 = 0 \]
- This gives the point \((-2, 0)\).

3. **Intersection of \(y = -2\) and \(y = -x - 2\)**:
- Substitute \(y = -2\) into \(y = -x - 2\):
\[ -2 = -x - 2 \implies -2 + 2 = -x \implies 0 = -x \implies x = 0 \]
- This gives the point \((0, -2)\).

### Step 4: Identify the vertices of the triangle
The vertices of the triangle formed by the lines are:
1. \((-2, -2)\)
2. \((-2, 0)\)
3. \((0, -2)\)

### Step 5: Find the circumcenter
Since the triangle formed by these points is a right triangle (the right angle is at \((-2, -2)\)), the circumcenter is located at the midpoint of the hypotenuse.

The hypotenuse is the line segment connecting the points \((-2, 0)\) and \((0, -2)\).

To find the midpoint \(M\) of the hypotenuse:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{-2 + 0}{2}, \frac{0 - 2}{2}\right) = \left(\frac{-2}{2}, \frac{-2}{2}\right) = (-1, -1) \]

### Final Answer
Thus, the circumcenter of the triangle is:
\[ \boxed{(-1, -1)} \]