Let's analyze the problem step by step:
1. Understanding Reflection Over the Line [tex]\(y = x\)[/tex]:
- When reflecting a point over the line [tex]\(y = x\)[/tex], the x-coordinates and y-coordinates of the point swap places.
- That is, for any point [tex]\((x, y)\)[/tex], its reflection over [tex]\(y = x\)[/tex] will be [tex]\((y, x)\)[/tex].
2. Identifying the Coordinates of Point [tex]\(D\)[/tex]:
- The given coordinates for point [tex]\(D\)[/tex] are [tex]\((a, b)\)[/tex].
3. Finding the New Coordinates After Reflection:
- According to the reflection rule, to find the new coordinates of the reflected point [tex]\(D^{\prime}\)[/tex], we swap [tex]\(a\)[/tex] (the x-coordinate) and [tex]\(b\)[/tex] (the y-coordinate).
- So, the coordinates of point [tex]\(D^{\prime}\)[/tex], the image after reflection, will be [tex]\((b, a)\)[/tex].
4. Choosing the Correct Option:
- We find that the correct coordinates of the image [tex]\(D^{\prime}\)[/tex] are [tex]\((b, a)\)[/tex].
Thus, the coordinates of the image [tex]\(D^{\prime}\)[/tex] after reflecting over the line [tex]\(y = x\)[/tex] are:
[tex]\[
\boxed{(b, a)}
\][/tex]
