Let x, y, z be complex numbers such that $xy$ z   9 yz zx xy x2 y2 z2    64 yz zx xy x3 y3 z3    488 yz zx xy $xy$ z $m$ If    where m, $n$ are positive integers with GCD (m, n) = 1, find $m$ + n. yz zx xy n

Step 1:   9

Step 2: yz zx xy

Step 3: 1 1 1

Step 4: x  $y$  z    12

Step 5: y z z $x$ x  y

Step 6: Let $x$ + $y$ + z = S1

Step 7: 1 1 1 12

Step 8:   

Step 9: y  z z  $x$ x  $y$ S1

Step 10: $xy$ z

Step 11: Now,  $x$  $y$  z     9S1

Step 12: y z z $x$ x y

Step 13: 64 + S1 = 9S1

Step 14: => S1 = 8

Step 15: x2 y2 z2

Step 16: x  $y$  z    64 S1

Step 17: y z z $x$ x y

Step 18: 488 + (x2 + y2 + z2) = 64 × 8

Step 19: x2 + y2 + z2 = 24

Step 20: xy + yz + zx = = 20

Step 21: 1 1 1 12 3

Step 22: Now,    

Step 23: xy yz zx 8 2

Step 24: 1 1 1 3

Step 25: 8–x 8–y 8–z 2

Step 26: (8 – x) (8 – y) (8 – z) = 56

Step 27: => xyz = 104

Step 28: x $y$ z 24 3 m

Step 29: So,     

Step 30: yz zx xy 104 13 n

Step 31: => $m$ + $n$ = 16