Born from a severed neck
Chrysaor enters the world at the moment of his mother’s death. Medusa lay with Poseidon in a soft meadow amid spring flowers, and the children of that union appear only when Perseus cuts off her head. Hesiod names them both in a single line: great Chrysaor and the horse Pegasus spring forth together from the severed neck. Hesiod explains the names in the lines that follow. Pegasus is so called because he was born near the springs of Ocean, for πηγαί are springs; and Chrysaor because he held a golden blade in his hands. The word ἄορ is the sword, and χρύσειον is golden: the name is a compound of the weapon he carries at the moment of his birth.
Two paths from one death
The brothers go separate ways at once. Pegasus flies up from the earth and comes to the gods, dwelling in the house of Zeus and bringing him the thunder and lightning. Chrysaor stays among mortals:
Χρυσάωρ δ᾽ ἔτεκεν τρικέφαλον Γηρυονῆα μιχθεὶς Καλλιρόῃ κούρῃ κλυτοῦ Ὠκεανοῖο
“But Chrysaor was joined in love to Callirrhoe, the daughter of glorious Ocean, and begot three-headed Geryones” — Hesiod, Theogony 287–288, tr. H. G. Evelyn-White
That is the whole of his story. He is born from Medusa’s neck with a golden blade in his hands, he marries an Oceanid, and he fathers a monster. Hesiod does not say he acts or speaks or rules. The Theogony lists him twice more — once in the genealogy of the gods, where Callirrhoe bears to stout-hearted Chrysaor a son who was the strongest of all men — but nothing is added to the picture. Geryon is the child the myth remembers, and Geryon is killed by Heracles in sea-girt Erythea for the sake of his shambling oxen. What Chrysaor does between his birth and his son’s death is a blank the poem does not fill.
The golden blade gives him his name, but no story says he uses it. Pegasus brings Zeus his thunder; Chrysaor begets a three-headed giant. The winged horse becomes a servant of the gods, and the man with the sword becomes the father of a monster Heracles will slay.