Golden Ellipse Calculus Level 5

Consider an ellipse whose semi-axes have lengths aa and bb, where {a>b}a>b. A chord in this ellipse makes acute angles \alphaα and \betaβ with the ellipse. Let \beta_{\text{min}}β
min
​
denote the minimum possible value of \betaβ, for a given value of \alphaα. Evaluate \beta_{\text{min}}β
min
​
as a function of \alphaα.

Now, take the ratio of the semi-axes of the ellipse to be \varphiφ (the golden ratio), and submit your answer as the value of {\displaystyle\int_{0}^{\frac{\pi}{ 2}}\tan\left(\beta_{\text{min}}\rig ht)\,d\alpha}∫
0
2
π
​

​
tan(β
min
​
)dα.