A conic section problem for year 2020

Put $a = \sqrt{2020}$ and $b = \sqrt{2019}$ , and let $m$ denote the slope of $l_1$ . The right focus of $E$ is $(1,0)$ , so the end points of $AB$ satisfy $y = m(x - 1) \text{ and } \frac {x^2}{a^2} + \frac {y^2}{b^2} = 1.$ Substituting $m(x - 1)$ for $y$ in the second equation gives $\frac {x^2}{a^2} + {m^2(x - 1)^2}{b^2} = 1,$ which is quadratic in $x$ and has solutions $\frac{a^2m^2 \pm ab \sqrt{(a^2 - 1)m^2 + b^2}}{a^2m^2 + b^2}.$ Notice that $a^2 - 1 = 2020 - 1 = 2019 = b^2$ , so these solutions can be written as $\frac{a^2m^2 \pm ab \sqrt{b^2m^2 + b^2}}{a^2m^2 + b^2} = \frac{a^2m^2 \pm ab^2 \sqrt{m^2 + 1}}{a^2m^2 + b^2}.$ Label these solutions $x_1$ and $x_2$ and put $y_1 = m(x_1 - 1)$ and $y_2 = m(x_2 - 1)$ . Then $AB = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} = \sqrt{(x_1 - x_2)^2 + m^2(x_1 - x_2)^2} = |x_1 - x_2| \sqrt{m^2 + 1} = \frac {2ab^2 (m^2 + 1)}{a^2 m^2 + b^2}.$ The slope of $l_2$ is $-m^{-1}$ . Repeating the algebra above using $-m^{-1}$ shows $CD = \frac {2ab^2(m^{-2} + 1)}{a^2m^{-2} + b^2} = \frac {2ab^2(m^2 + 1)}{a^2 + b^2 m^2}.$ Therefore, $d = AB + CD = \frac{2ab^2(m^2 + 1)}{a^2m^2 + b^2} + \frac{2ab^2(m^2 + 1)}{a^2m^2 + b^2} = \frac {2ab^2(a^2 + b^2)(m^2 + 1)^2}{(a^2m^2 + b^2)(a^2 + b^2m^2)}, \tag{1}$ and additionally $S = \frac 12 \cdot AB \cdot CD = \frac 12 \cdot \frac {2ab^2(m^2 + 1)}{a^2m^2 + b^2} \cdot \frac {2ab^2(m^2 + 1)}{a^2 + b^2m^2} = \frac {2a^2b^4(m^2 + 1)^2}{(a^2m^2 + b^2)(a^2 + b^2m^2)}. \tag{2}$ By (1) and (2), the extreme values of $d$ and $S$ depend on $\frac {(m^2 + 1)^2}{(a^2m^2 + b^2)(a^2 + b^2m^2)},$ and this value achieves its maximum when $m = 0$ and its minimum when $m = 1.$ Evaluating (1) and (2) at $m = 0$ and $m = 1$ shows $179.73313000 \approx \frac {8ab^2(a^2 + b^2)}{(a^2 + b^2)^2} \leq d \leq \frac {2ab^2(a^2 + b^2)}{a^2b^2} \approx 179.73314102$ and also $4037.99975248 \approx \frac {8a^2b^4}{(a^2 + b^2)^2} \leq S \leq 2b^2 = 4038.$ Hence the desired value is $\lfloor 10^7(d_{\text{max}} - d_{\text{min}} + S_{\text{max}} - S_{\text{min}}) \rfloor = \lfloor 10^7 (179.73314102 - 179.73313000 + 4038 - 4037.99975248) \rfloor = \boxed{2585}.$