Bot GPT 4-o 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:
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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.
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Find the slope: The line \(xy = x\) also represents a transformation in geometry that generally represents \(y\) about the line \(x = y\).
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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 \(y\).
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Reflect the point:
- The original point is \((-1, -2)\).
- Swap the coordinates: This transforms \((-1, -2)\) into \((-2, -1)\).
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Determine the location: Now, we perform the reflection over the line \(y = x\) (or \(xy = x\)) on \((-2, -1)\).
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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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