I'm not awfully familiar with polar coordinates, and I'm trying to prove Kepler's second law via observations. I found plenty of articles regarding proof of Kepler's second law, and the following seemed the easiest to understand: https://www.math.ksu.edu/~dbski/writings/planetary.pdf (third page).
My question is, how can I move $r$ so that it will stretch from a focal point $(c, 0)$ instead of $(0, 0)$? And when doing so, will/can it change the function in the following integral?
$$\text{Area swept between times}\ t_1\ \text{and}\ t_2=\frac12\int_{\theta_1}^{\theta_2}r^2\ \mathrm d\theta$$
Since $r=a-\dfrac{cx}a$, I figured moving the starting point of $r$ to $(c, 0)$ and rewriting $x$ to $\theta$ through trig would be the best way to implement my given information into the integral.
I have $a$, $b$ and $c$, which is why I want to hold onto that function.