Golden Ellipse

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

Now, take the ratio of the semi-axes of the ellipse to be $\varphi$ (the golden ratio), and submit your answer as the value of ${\displaystyle\int_{0}^{\frac{\pi}{2}}\tan\left(\beta_{\text{min}}\right)\,d\alpha}$ .