I'm solving a physics question where the nearest and furthest distance of a satellite from Earth are given, along with the satellite's speed at the nearest point, and it asks to find the period of the satellite's orbit. In solving the question, the answer uses the following equation for the area of an ellipse:
$$S=\frac{\pi}{2}(\overline{F_1V_1}+\overline{F_1V_2})\sqrt{\overline{F_1V_1}\cdot\overline{F_1V_2}}$$
I have never seen this equation before, and searches did not return any results.
Does anyone know a derivation for this equation?
Your description of $a,b$ doesn't match the diagram. Anyway, I think you don't want $F_1,F_2$ (which would make your $a=b$) but a single focus $F$ and measure the distances $V_1,V_2$ from it, say $\ell_1,\ell_2$.
Then $\ell_1+\ell_2=2a$ and $\ell_1\ell_2=(a+OF)(a-OF)=b^2$ and we recover the usual formula for the area $\pi ab$ of an ellipse with semimajor and semiminor axes $a,b$.