בראשית כד:ב פירוש: רש״י, רשב״ם, רד״ק וטור הארוך

זקן ביתו THE ELDER OF HIS HOUSE — Because the noun is in the construct state the ק is punctuated with a Patach.

PUT YOUR HAND UNDER MY THIGH. And similarly with Yosef when he swore to Ya'akov. The making of a covenant or an oath of a son, or of a slave to his master was in this fashion, for they shared the aspect of subjugation, and it is thus written (Mal. 1:6), "A son should honor his father, and a slave his master." However, handshaking or cutting an object in two for a covenant and passing between the pieces is found with other persons.

ויאמר...זקן ביתו, the word zekan is a genitive (adjective) of the word zaken just as the word kevod is a genitive of the word kavod. Its meaning here is that this servant had grown up in Avraham’s household until he himself had become elderly. Or, its meaning could be that this servant was older (in the sense of senior) to all Avraham’s other servants.