What is the ruling on getting rid of two old cats, that have been living in the house for eight years, and are causing hardship in caring for them and purifying the house from their impurity, given their owner's fear of them being killed in an animal shelter or dying in the street, and his fear of the consequences of his action, remembering the Hadith of the woman who entered Hellfire because of a cat?

The urine and droppings of cats are considered impure (najis) by the majority of scholars, and the prayer area must be purified from them. To get rid of cats, they can be taken out to forests outside the city, and there is no sin in this, as Allah Almighty says: "And how many a creature there is that does not carry its [own] provision. Allah provides for it and for you." The woman mentioned in the Hadith (definition) was punished because she imprisoned a cat until it died of hunger, not merely for leaving it.