(In reply to alternate solution
"1/p^2 + 1/q^2 + 1/r^2 + 6 = 0"
....... and since each of p, q and r must be non zero real numbers, the lhs of the above equation must be > 6, which leads to a contradiction.
Very nice methodology especially since the said process obviates the necessity to consider separately the positive case and the negative case.
Consequently, I confirm having hyperlinked your methodology in my solution.
Edited on May 5, 2008, 2:50 pm