192 Optimizing Optimization
Simplifying , and letting Δ α γ β 2 0,vvvvv v^22222
2212 2 2 2
210()
()γθ βθ βθ β θ βθ
βθ αθγθ
(8.74)or,
vv(^2) γβαθ 20 ()Δ
(8.75)^
So that
v
β
γ
βαγ θγ
γ
(^2) Δ
(8.76)
Since ,
βαγ θγ
(^2) Δ 0
(8.77)^
or
ΔΔθγ^0 (8.78)^
or
Δ()θγ−^10 (8.79)^
so that
θγ^1 (8.80)^
The solution for the upper part of the hyperbola is where v
β
γ
and this
corresponds to:
v
β
γ
βαγ θγ
γ
(^2) Δ
(8.81)