Consider the circle $x^2 + y^2 = a^2$, where a is any positive number. Let $\mathscr L$ be the line y = b where b is a real number with |b| < a. The y-axis and $\mathscr L$ divide the circle into four regions. Suppose the area of the upper right region is $R_1$ and going counter-clockwise, the other regions have areas $R_2$, $R_3$, and $R_4$. If $X = R_1 - R_4 + R_3 - R_2$, what is true of $X$?
This is not a homework question, it's a question from a mathleague competition that I did not understand. Please keep answers at the high school mathematics level.
The circle is at the origin, hence symmetric along y-axis and y divides $C$ into two equal regions $R_{l}$ and $R_{r}$.
Let $\mathcal{L}$ divide $R_{l}$ into $R_{2} = R_{lt}$, $R_3 = R_{lb}$, and $R_{r}$ into $R_1 = R_{rt}, R_4 = R_{rb}$.
By symmetry $R_1 = R_2$ and $R_3 = R_4$, therefore $R_1 - R_4 = R_2 - R_3$ and $x = 0$.