Is an Aqiqah sufficient with a ewe that was designated for it, then she became pregnant and gave birth, and the Aqiqah was performed with her alone without her offspring? And is it obligatory for him to perform the Aqiqah with the ewe and her offspring, or with one of them?

It is permissible for the 'aqiqah to be performed with the mother sheep, and her offspring can be sacrificed as a dependency to her, because the ruling on 'aqiqah is like the ruling on udhiyah (sacrificial animal). The jurists have stipulated that it is desirable to slaughter the offspring of the udhiyah along with it, whether the mother was pregnant when designated or became pregnant afterwards, because the offspring becomes an udhiyah as a dependency to its mother.