Let $a,b,c,p,q$  be real numbers. Suppose  $\alpha ,\beta$ are the roots of the equation $x^2+2px+q=0$  and $\alpha ,1/\beta$ are the roots of the equation $a{x}^2+2bx+c=0$ , where  ${\beta }^2\notin \{-1,0,1\}$
Statement-1:  $(p^2-q)\ge 0,(b^2-ac)\ge 0$
Statement-2:  $b\ne pa\text{ or }c\ne qa$
 

Suppose the roots are imaginary. Then 
 $\beta =\overline{\alpha }\text{ and }\frac{1}{\beta }=\overline{\alpha }\Rightarrow \beta =\frac{1}{\beta }\Rightarrow {\beta }^2=1$
Which is not possible. The roots are real, so 
 $(p^2-q)\ge 0,(b^2-ac)\ge 0$
Hence, statement 1 is correct.
Also, $-2b/a=\alpha +\beta$  and  $\alpha \beta =c/a,\alpha +\beta =-2p,\alpha \beta =q$
If  $\beta =1,\text{\hspace{0.17em}then}$
$\alpha =q\Rightarrow c=qa$ (which is not possible)
Also, $\alpha +1=\frac{-2b}{a}\Rightarrow -2p=\frac{-2b}{a}\Rightarrow b=ap$
 (which is not possible)
Hence, statement 2 is correct, but it is not correct explanation of statement 1