Let z be a unit complex number having the argument θ, 0<θ<π2 and satisfying the relation |z−3i|=3, then arg(cotθ−6z)
A
π4
No worries! We‘ve got your back. Try BYJU‘S free classes today!
B
π3
No worries! We‘ve got your back. Try BYJU‘S free classes today!
C
π2
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
D
3π4
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution
The correct option is Cπ2 From polar form of complex numbers, z=|z|(cosθ+isinθ) As z is uni modular, z=cosθ+isinθ We have |z−3i|=3 ⇒|cosθ+i(sinθ−3)|=3 ⇒√cos2θ+(sinθ−3)2=3 ⇒cos2θ+(sinθ−3)2=9 ⇒cos2θ+sin2θ−6sinθ+9=9 ⇒1−6sinθ=0 ⇒sinθ=16…(1)
Now, cotθ−6z=cotθ−6(cosθ+isinθ) =cotθ−6(cosθ−isinθ) =cotθ−6sinθ(cotθ−i) By equation (1) =cotθ−cotθ+i=i Therefore, arg(cotθ−6z)=arg(i)=π2