Question
What is the image of left parenthesis, minus, 1, comma, minus, 2, right parenthesis(−1,−2) after a reflection over the line y, equals, xy=x?
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Answered by
GPT-4o mini
To find the image of the point \((-1, -2)\) after reflecting it over the line \(xy = x\), we need to proceed with the following steps:
1. **Identify the line of reflection**: The equation \(xy = x\) can be rearranged to \(y = 1\) (for \(x \neq 0\)). This line can be interpreted as the reflection line. However, it often describes a 45-degree line in the coordinate plane.
2. **Find the slope**: The line \(xy = x\) also represents a transformation in geometry that generally represents \(y\) about the line \(x = y\).
3. **Convert the original coordinates**: Given point \((-1, -2)\), we consider its coordinates in relation to the line \(y = x\). The transformation involves swapping \(x\) and \(y\).
4. **Reflect the point**:
- The original point is \((-1, -2)\).
- Swap the coordinates: This transforms \((-1, -2)\) into \((-2, -1)\).
5. **Determine the location**: Now, we perform the reflection over the line \(y = x\) (or \(xy = x\)) on \((-2, -1)\).
6. **Final result**: The reflection over the line \(xy = x\) (which is equivalent to the transformation swap) takes the point to \((-1, -2)\) to \((-2, -1)\).
Thus, the final output shows the new reflected coordinates:
\[
\boxed{(-2, -1)}
\]
1. **Identify the line of reflection**: The equation \(xy = x\) can be rearranged to \(y = 1\) (for \(x \neq 0\)). This line can be interpreted as the reflection line. However, it often describes a 45-degree line in the coordinate plane.
2. **Find the slope**: The line \(xy = x\) also represents a transformation in geometry that generally represents \(y\) about the line \(x = y\).
3. **Convert the original coordinates**: Given point \((-1, -2)\), we consider its coordinates in relation to the line \(y = x\). The transformation involves swapping \(x\) and \(y\).
4. **Reflect the point**:
- The original point is \((-1, -2)\).
- Swap the coordinates: This transforms \((-1, -2)\) into \((-2, -1)\).
5. **Determine the location**: Now, we perform the reflection over the line \(y = x\) (or \(xy = x\)) on \((-2, -1)\).
6. **Final result**: The reflection over the line \(xy = x\) (which is equivalent to the transformation swap) takes the point to \((-1, -2)\) to \((-2, -1)\).
Thus, the final output shows the new reflected coordinates:
\[
\boxed{(-2, -1)}
\]
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