Thus the expression has a maximum of 135 when θ=−ϕ.
Alternatively, by the Cauchy-Schwarz Inequality,
81cosθ−108sinθ≤(cos2θ+sin2θ)(812+1082)=135,
with equality occurring when sinθ=−135108=−54. Hence the requested greatest real part is 4⋅135=540.
OR
As in the first solution,
z1=∣z∣2zˉ=16zˉ,
so the given expression can be written as (75+117i)z+(6+9i)zˉ. The real part of a complex number is not affected by conjugation, so the real part the given expression is the same as the real part of
As z varies on the circle of radius 4 centered at the origin, the number (81+108i)z rotates around a circle centered at the origin with radius ∣z∣⋅812+1082=4⋅992+122=540. Thus the given expression has a maximum real part for the value of z corresponding to the point of this circle on the positive real axis, and that real part is 540.