If X is the center of dilation, the only lines that go through X after the dilation are the ones that go through X before the dilation: XY (and XY'), XZ (and XZ').
Line AY does not go through X, so A'Y' will not go through X. Rather, the line will be moved a distance from X according to the dilation factor. The line A'Y' will be parallel to AY.
