The union of open discs $C_n$ in $\mathbb{R}^2$ centered at $(n,0)$ with radius $n$ 
For  each $n\ge 1$, let $C_{n}$ be  the  open  disc in $\mathbb{R}^2$, with  centre at the  point $(n,0)$ and  radius equal  to $n$. Then   $\mathcal{C} = \cup C_{n}$ is 
  
  
*
  
*$\{(x,y)\in\mathbb{R}^2 : x\gt 0 \text{ and } |y|\lt x \}$
  
*$\{(x,y)\in\mathbb{R}^2 : x\gt 0 \text{ and } |y|\lt 2x\}$
  
*$\{(x,y)\in\mathbb{R}^2 : x\gt 0 \text{ and } |y|\lt 3x\}$
  
*$\{(x,y)\in\mathbb{R}^2 : x\gt 0 \}$
  

Now I  drew  this  picture

Drawing two  diameters  of  the  circle  $C_{8}$ with  green  line I  think the  largest value  of $|y|$  for  any point in $C_{8}$  is $8$  but for  the  same  points $2x$ is $16$. So for  every  circle the relation  $|y|<2x$  will  hold  for  the  points  in  that  circle. Then that  makes option $2$. correct. But  the given  answer  is  option $4$. So  please  help!
 A: Pick any point in $\{(x,y)\in\mathbb{R}^2 : x\gt 0 \}$ and you can find a circle that contains it. Pick any point not in $\{(x,y)\in\mathbb{R}^2 : x\gt 0 \}$ and you can not find a circle that encloses it. This is not true for the other options.
In other words, $C$ is the union of all the open disks which grow unbounded to the right. So their union will be the entire right half of the $\mathbb{R}^2$. 
A: Hint.
Take a point $P=(x,y)$ of the plane with $x>0$. Can you find a circle $C$ centered on the $x$-axis at point $(a,0)$ for which $P \in C$?
Now what do you think of the position of $P$ vs. a disk centered at $(n,0)$ of radius $n$ with $n$ integer and $n >a$?
Answering those questions will lead you to response 4.
A: It is enough to consider the point $(1, 4)$, which lies in the fourth set but not the other three. We want to find some $n \in \mathbb N$ such that:
$$
(1, 4) \in C_n = \{(x, y) \in \mathbb R^2 \mid (x - n)^2 + y^2 < n^2\}
$$
Plugging in the numbers and working through the algebra, we get:
$$
(1 - n)^2 + 4^2 < n^2
\iff (n^2 - 2n + 1) + 16 < n^2
\iff 17 < 2n
\iff n > 17/2 = 8.5
$$
Hence, it follows that $(1, 4) \in C_9, C_{10}, C_{11},$ and so on.
