If ∣∣∣z−4z]=2, Then the maximum value of |z| is equal tp
Consider the following question.
Z=ρeiϕ
We know that.
|Z|=√Z¯¯¯¯Z
(ρeiϕ−4ρe−iϕ)(ρe−iϕ−4ρeiϕ)=4
ρ2−8cos(2ϕ)+16ρ2=4
ρ4−2(2+4cos(2ϕ))ρ2+16=0
Now solving for ρ2
ρ2=2(1+2cos(2ϕ)±√4cos(2ϕ)+2cos(4ϕ)−1)
ρ2max=2(1+2+√4+2−1)=2(3+√5)
max|Z|=√2(3+√5)=1+√5
Hence, this is the required answer.