Considering the equation of the circle $x^2+y^2=9$ and a line $y=1$ which divides the circle into two regions let small region be $R_1$ and other be $R_2$. Which of the following statements is/are CORRECT?

A) 

$$\begin{gathered} C{C}_1=3-R \\ {\alpha }^2+{\beta }^2={(3-R)}^2 \\ (\beta -1)=R\Rightarrow \beta =1+R \\ {\alpha }^2={(3-R)}^2-{(1+R)}^2 \\ =(9+R^2-6R)-(R^2+1+2R) \\ {\alpha }^2=8-8R\Rightarrow R=1 then centre lies on y –axis \end{gathered}$$

B)  
$$\begin{gathered} C_1{C}_2=3-R_2 \\ 1-\beta =R\Rightarrow \beta =1-R \\ {\alpha }^2+{\beta }^2={(3-R)}^2 \\ \Rightarrow {\alpha }^2={(3-R)}^2-{(1-R)}^2 \\ = 8 – 4R \\ \Rightarrow R=2 then centre lies on y-axis \end{gathered}$$

C)   
 

$$\begin{gathered} R^2+{(1-R)}^2={(3-R)}^2 \\ {R}^2+R^2+1-2R=9+R^2-6R \\ \Rightarrow R^2+4R=8 \\ \Rightarrow R^2+4R+4=12 \\ \Rightarrow {(R+2)}^2=12 \\ \Rightarrow R=2\sqrt{3}-2=2(\sqrt{3}-1) \end{gathered}$$

D)  

$$\begin{gathered} 2\sqrt{R^2-d^2}=2\sqrt{5} \\ \Rightarrow R^2-d^2=5 \\ \Rightarrow R^2-1=5\Rightarrow R^2=6 \\ \Rightarrow R=\sqrt{6} \end{gathered}$$