All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
Cats are not impure. Kabshah bint Ka‘b ibn Maalik, the wife of Abu Qatadah's son, may Allaah be Pleased with them, said:
"Abu Qatadah visited (me) and I poured out water for him for Wudhoo’ (ablution). A cat came and drank some of it and he tilted the vessel for it until it drank its full. He (Abu Qatadah) saw me looking at him and he asked, "Are you surprised my niece?" I said, "Yes." He said that the Messenger of Allaah said, "It is not impure; it is one of those (males or females) who go round among you." [Abu Dawood, An-Nasa’i, At-Tirmithi and Ibn Majah]
Consequently, the fallen hair of a cat is pure and does not render food or drink impure. However, one should remove what is possible to remove of it in order to preserve one's health.
Allah Knows best.