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 equations of the lines
First, we need to rewrite the given equations in a more manageable form.

1. The first equation is \(xy + 2x + 2y + 4 = 0\). We can rearrange this as:
\[ xy + 2x + 2y = -4 \]
This can be factored as:
\[ (x + 2)(y + 2) = 0 \]
This gives us two lines:
\[ x + 2 = 0 \quad \text{(or)} \quad y + 2 = 0 \]
Thus, the lines are \(x = -2\) and \(y = -2\).

2. The second equation is \(x + y + 2 = 0\), which can be rearranged to:
\[ y = -x - 2 \]

### Step 2: Find the points of intersection
Next, we will find the points of intersection of these lines to form the vertices of the triangle.

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

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

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

### Step 3: Identify the vertices of the triangle
Now we have the vertices of the triangle:
- \(A(-2, -2)\)
- \(B(-2, 0)\)
- \(C(0, -2)\)

### Step 4: Determine the circumcenter
Since triangle \(ABC\) is a right triangle (with the right angle at \(C\)), the circumcenter is located at the midpoint of the hypotenuse \(AB\).

1. **Calculate the midpoint of \(AB\)**:
- The coordinates of \(A\) and \(B\) are:
\[ A(-2, -2) \quad \text{and} \quad B(-2, 0) \]
- The midpoint \(M\) of \(AB\) is given by:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{-2 + (-2)}{2}, \frac{-2 + 0}{2}\right) \]
- This simplifies to:
\[ M = \left(-2, -1\right) \]

### Final Answer
Thus, the circumcenter of the triangle formed by the given lines is:
\[ \text{Circumcenter} = (-2, -1) \]