Pointdrips are drips coming down little by little. These are the drips of an intravenous drip. Back in the day this was a way of talking about raindrops, or really any falling drops of water. But nowadays we only use it for IV drips.
This is a jukugo word that uses the on'yomi readings of the kanji. You should be able to read this on your own.
After getting an IV drip, he was thoroughly examined.