Question Number 102598 by dw last updated on 10/Jul/20

Answered by mr W last updated on 10/Jul/20

z=2(cos θ+i sin θ)  ∣z−i∣=(√(4cos^2  θ+(2 sin θ−1)^2 ))=(√(5−4 sin θ))  ∣z+i∣=(√(4cos^2  θ+(2 sin θ+1)^2 ))=(√(5+4 sin θ))  A=((∣z−i∣)/(∣z+i∣))=(√((5−4 sin θ)/(5+4 sin θ)))=(√(((10)/(5+4 sin θ))−1))  maximum is if sin θ=−1,  A_(max) =(√(((10)/(5−4))−1))=3  minimum is if sin θ=1,  A_(min) =(√(((10)/(5+4))−1))=(1/3)

Answered by Dwaipayan Shikari last updated on 10/Jul/20

((∣z−i∣)/(∣z+i∣))=∣((z−i)/(z+i))∣=a    {for maximum value  ∣((z+i)/(z−i))∣=(1/a)  ∣(z/i)∣=((1+a)/(1−a))⇒∣z∣=((1+a)/(1−a))=2  so    a=(1/3)