The coordinates of the image of the origin O with respect to the line `x+y+1=0` are

Options

A. `(-1/2,-1/2)`
B. `(-2,-2)`
C. `(1,1)`
D. `(-1,-1)`

Answer

To find the coordinates of the image of the origin \( O(0, 0) \) with respect to the line \( x + y + 1 = 0 \), we can use the formula for finding the image of a point with respect to a line.

### Step-by-step solution:

1. **Identify the line equation**: The given line is \( x + y + 1 = 0 \). We can rewrite it in the standard form \( ax + by + c = 0 \) where:
- \( a = 1 \)
- \( b = 1 \)
- \( c = 1 \)

2. **Identify the point**: The origin \( O \) has coordinates \( (x_1, y_1) = (0, 0) \).

3. **Use the image formula**: The formula for the coordinates of the image \( (x_2, y_2) \) of a point \( (x_1, y_1) \) with respect to the line \( ax + by + c = 0 \) is given by:
\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = -\frac{2(ax_1 + by_1 + c)}{a^2 + b^2} \]

4. **Substitute the values**:
- Here, \( x_1 = 0 \), \( y_1 = 0 \), \( a = 1 \), \( b = 1 \), \( c = 1 \).
- Substitute these into the formula:
\[ \frac{x - 0}{1} = \frac{y - 0}{1} = -\frac{2(1 \cdot 0 + 1 \cdot 0 + 1)}{1^2 + 1^2} \]
\[ \frac{x}{1} = \frac{y}{1} = -\frac{2(0 + 0 + 1)}{1 + 1} \]
\[ \frac{x}{1} = \frac{y}{1} = -\frac{2}{2} \]
\[ x = y = -1 \]

5. **Conclusion**: The coordinates of the image of the origin \( O \) with respect to the line \( x + y + 1 = 0 \) are \( (-1, -1) \).

### Final Answer:
The coordinates of the image of the origin \( O \) with respect to the line \( x + y + 1 = 0 \) are \( (-1, -1) \).