已知 $P_{1}\left(x_{1}, y_{1}\right), P_{2}\left(x_{2}, y_{2}\right)$ 是抛物线 $y=a x^{2}-2 a x$ 上的点, 下列命题正确的是 ( )
A. 若 $\left|x_{1}-1\right|>\left|x_{2}-1\right|$, 则 $y_{1}>y_{2}$
B. 若 $\left|x_{1}-1\right|>\left|x_{2}-1\right|$, 则 $y_{1} < y_{2}$
C. 若 $\left|x_{1}-1\right|=\left|x_{2}-1\right|$, 则 $y_{1}=y_{2}$
D. 若 $y_{1}=y_{2}$, 则 $x_{1}=x_{2}$