Circles of radius $2$ passing through origin with centers on $x=1$ 
There are two circles of radius $2$ that have centers on the line $x=1$
  and pass through the origin. Find their equations.

Please explain to me what the problem is really saying.

A circle with centre $(a,b)$ and radius $r>0$ has equation $(x-a)^2+(y-b)^2=r^2$. Since these circles have centre on the line $x=1$ and radius $2$, they have equation $(x-1)^2+(y-b)^2=4$. Since the circles pass through the point $(0,0)$, we have $(0-1)^2+(0-b)^2=4$, so $b^2+1=4$. Solving for $b$ gives $b=\pm\sqrt 3$, so the circles are given by $(x-1)^2+(y-\sqrt 3)^2=4$ and $(x-1)^2+(y+\sqrt 3)^2=4$.
