Why did the wise woman ask the king to call to mind Yahweh his God?

The wise woman asked the king this so that the avenger of blood would not destroy anyone further, so that they would not destroy her son.

Who did the king swear by when he told the woman that not one hair of her son would fall to the ground?

The king swore by Yahweh by saying, “As Yahweh lives, not one hair of your son will fall to the ground.”