If z is a complex number lying in the fourth quadrant of the Argand plane and ` |(kz)/(k+1) + 2i|gt sqrt2` for all real values of ` k ( k ne -1)` the find the range of arg (z)

Explanation

The given inequality is ` |(kz)/(k+1) + 2i|gt sqrt2`
`Rightarrow |z-((-2i)(k+1))/k|gt(sqrt2(k+1))/k`
Which represents the region lying outside the circle centred
`at(0,(-2(k+1))/k) and " radius " (sqrt2(k+1))/k`
let `COP =theta` so that
`sin theta= (CP)/(OC) = ((sqrt2(k+1))/k)/((2(k+1))/k)=1/sqrt2 = sinpi//4`
` theta= pi/4`
Since z lies in the `IV^("th")` quadrant, hence
` arg z "in" ( - pi/4, 0)`