If z 1 and z 2 are two distinct complex numbers satisfying the relation z 1 2 - z 2 2 = z - 1 2 + z - 2 2 - 2 z - 1 z - 2 and arg ⁡ z 1 - arg ⁡ z 2 = a π b , then the least possible value of a - b is equal to (where, a & b are integers)

Question

If z 1 and z 2 are two distinct complex numbers satisfying the relation z 1 2 - z 2 2 = z - 1 2 + z - 2 2 - 2 z - 1 z - 2 and arg ⁡ z 1 - arg ⁡ z 2 = a π b , then the least possible value of a - b is equal to (where, a & b are integers),

MARKS App · Accepted Answer

z 1 2 - z 2 2 = z - 1 2 + z - 2 2 - 2 z - 1 z - 2 z 1 2 - z 2 2 = z 1 2 + z 2 2 - 2 z 1 z 2 z 1 + z 2 z 1 - z 2 = z 1 - z 2 z 1 - z 2 ⇒ z 1 + z 2 = z 1 - z 2 z 1 ¯ ⊥ z 2 ¯ ⇒ arg ⁡ z 1 z 2 = 2 n π ± π 2 i.e. - π 2 , π 2 , 3 π 2 , … … . . The minimum value of a - b = 1 (when a = 1 & b = 2 )

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