You could first read in the first line like so:

open FILE, "<file.txt" or die "Could not open file.txt: $!";
my $line = <FILE>;
close FILE;
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I am not sure what you are trying to grep here, but your unwanted matches are due to two reasons: you didn't escape the metacharacter `.' and you probably didn't exclude stuff off the end of the string. Try something like, e.g.,

$line =~ m|^\Q$path/$file\E$|;
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The ^ and $ make the regexp match just the exact string and nothing else on the line, and the \Q...\E quote any metacharacters in your string.

-Mark