Does passive perception supersede active perception?


If a player makes an active perception check and it is lower than their passive perception, should we use the passive perception instead?

Background:
During our weekly game yesterday the party scout was actively searching for hidden monsters. He rolled poorly and ended up with a 7 on his perception check, seeing nothing. The claim was then made that since his Passive Perception was a 15, he would intuitively notice anything that he didn’t actively spot.