The quadrilatera formed by the pair of lines `xy+x+y+1=0,xy+3x+3y+9=0` is

Options

A. parallelogram
B. rhombus
C. rectangle
D. square

Answer

To solve the problem of finding the quadrilateral formed by the pair of lines given by the equations \(xy + x + y + 1 = 0\) and \(xy + 3x + 3y + 9 = 0\), we will follow these steps:

### Step 1: Factor the first equation
The first equation is:
\[ xy + x + y + 1 = 0 \]
We can rearrange this as:
\[ xy + x + y = -1 \]
Now, we can factor it:
\[ y(x + 1) + (x + 1) = 0 \]
This gives us:
\[ (x + 1)(y + 1) = 0 \]
From this, we can find the lines:
1. \(x + 1 = 0 \Rightarrow x = -1\)
2. \(y + 1 = 0 \Rightarrow y = -1\)

### Step 2: Factor the second equation
The second equation is:
\[ xy + 3x + 3y + 9 = 0 \]
Rearranging gives:
\[ xy + 3x + 3y = -9 \]
Factoring this, we have:
\[ y(x + 3) + 3(x + 3) = 0 \]
This simplifies to:
\[ (x + 3)(y + 3) = 0 \]
From this, we can find the lines:
1. \(x + 3 = 0 \Rightarrow x = -3\)
2. \(y + 3 = 0 \Rightarrow y = -3\)

### Step 3: Identify the intersection points
Now we have the following lines:
1. \(x = -1\)
2. \(y = -1\)
3. \(x = -3\)
4. \(y = -3\)

The intersection points of these lines are:
- Intersection of \(x = -1\) and \(y = -1\) gives the point \((-1, -1)\)
- Intersection of \(x = -1\) and \(y = -3\) gives the point \((-1, -3)\)
- Intersection of \(x = -3\) and \(y = -1\) gives the point \((-3, -1)\)
- Intersection of \(x = -3\) and \(y = -3\) gives the point \((-3, -3)\)

### Step 4: Determine the shape of the quadrilateral
The points we found are:
1. \((-1, -1)\)
2. \((-1, -3)\)
3. \((-3, -3)\)
4. \((-3, -1)\)

Plotting these points, we can see that they form a square:
- The distance between \((-1, -1)\) and \((-1, -3)\) is \(2\) units (vertical distance).
- The distance between \((-1, -1)\) and \((-3, -1)\) is \(2\) units (horizontal distance).

Since all sides are equal and the angles are right angles, the quadrilateral is indeed a square.

### Conclusion
The quadrilateral formed by the pair of lines is a square with side length \(2\) units.

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