After getting an IV drip, he was thoroughly examined.
Point drips 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.