Answer :
To determine which reflection of the point \((0, k)\) will produce an image at the same coordinates, we need to consider the effects of each reflection on the point \((0, k)\).
1. Reflection across the \(x\)-axis:
- When reflecting a point \((x, y)\) across the \(x\)-axis, its coordinates change to \((x, -y)\).
- For the point \((0, k)\), reflecting it across the \(x\)-axis results in \((0, -k)\).
- This is a different point from \((0, k)\).
2. Reflection across the \(y\)-axis:
- When reflecting a point \((x, y)\) across the \(y\)-axis, its coordinates change to \((-x, y)\).
- For the point \((0, k)\), reflecting it across the \(y\)-axis results in \((0, k)\).
- This is the same point as the original one.
3. Reflection across the line \(y=x\):
- When reflecting a point \((x, y)\) across the line \(y=x\), its coordinates switch to \((y, x)\).
- For the point \((0, k)\), reflecting it across the line \(y=x\) results in \((k, 0)\).
- This is a different point from \((0, k)\).
4. Reflection across the line \(y=-x\):
- When reflecting a point \((x, y)\) across the line \(y=-x\), its coordinates switch and change sign to \((-y, -x)\).
- For the point \((0, k)\), reflecting it across the line \(y=-x\) results in \((-k, 0)\).
- This is a different point from \((0, k)\).
From the above analysis, the reflection that results in the image remaining at \((0, k)\) is the reflection across the \(y\)-axis. Thus, the correct answer is:
Reflection across the [tex]\(y\)[/tex]-axis.
1. Reflection across the \(x\)-axis:
- When reflecting a point \((x, y)\) across the \(x\)-axis, its coordinates change to \((x, -y)\).
- For the point \((0, k)\), reflecting it across the \(x\)-axis results in \((0, -k)\).
- This is a different point from \((0, k)\).
2. Reflection across the \(y\)-axis:
- When reflecting a point \((x, y)\) across the \(y\)-axis, its coordinates change to \((-x, y)\).
- For the point \((0, k)\), reflecting it across the \(y\)-axis results in \((0, k)\).
- This is the same point as the original one.
3. Reflection across the line \(y=x\):
- When reflecting a point \((x, y)\) across the line \(y=x\), its coordinates switch to \((y, x)\).
- For the point \((0, k)\), reflecting it across the line \(y=x\) results in \((k, 0)\).
- This is a different point from \((0, k)\).
4. Reflection across the line \(y=-x\):
- When reflecting a point \((x, y)\) across the line \(y=-x\), its coordinates switch and change sign to \((-y, -x)\).
- For the point \((0, k)\), reflecting it across the line \(y=-x\) results in \((-k, 0)\).
- This is a different point from \((0, k)\).
From the above analysis, the reflection that results in the image remaining at \((0, k)\) is the reflection across the \(y\)-axis. Thus, the correct answer is:
Reflection across the [tex]\(y\)[/tex]-axis.
