Let z₁ and z₂ be two complex numbers such that | z₁+z₂ z₁-z₂ |=1 , then what is Re ( z₁ z₂ )+1 equal to?

Step-by-step solution

Let z₁=x₁+i y₁ and z₂=x₂+i y₂ Now, | z₁+z₂ z₁-z₂ |=1 aligned & |z₁+z₂ |= |z₁-z₂ | & | (x₁+x₂ )+i (y₁+y₂ ) |= (x₁-x₂ ) & +i (y₁-y₂ ) & (x₁+x₂ )^2+ (y₁+y₂ )^2= (x₁-x₂ )^2 & + (y₁-y₂ )^2 & x₁^2+x₂^2+2 x₁ x₂+y₁^2+y₂^2+2 y₁ y₂ & =x₁^2+x₂^2-2 x₁ x₂+y₁^2 & +y₂^2-2 y₁ y₂ & x₁ x₂+y₁ y₂=0 & z₁ z₂ = x₁+i y₁ x₂+i y₂ x₂-i y₂ x₂-i y₂ & = (x₁ x₂+y₁ y₂ )+i (y₁ x₂-x₁ y₂ ) x₂^2+y₂^2 & =0+ i (y₁ x₂-x₁ y₂ ) x₂^2+y₂^2 aligned aligned & Re ( z₁ z₂ )=0 & Re ( z₁ z₂ )+1=1 . aligned