Thank you for reporting, we will resolve it shortly
Q.
If the curve $y=a x^2+2 b x+c(a, b, c \in R, a, c \neq 0)$ never meets the $x$-axis, then $a, b, c$ can be in
Sequences and Series
Solution:
$ D < 0 \Rightarrow 4 b ^2-4 ac < 0$
$b ^2- a c <0$
(i) If $b=\frac{a+c}{2} \Rightarrow(a+c)^2-4 a c< 0 \Rightarrow(a-c)^2< 0$
(ii)$b ^2= ac $
$\therefore 0<0$
(iii)$b=\frac{2 a c}{a+c} $
$\therefore \frac{4 a^2 c^2}{(a+c)^2}-a c< 0$
$\frac{a c}{(a+c)^2}\left[4 a c-(a+c)^2\right]< 0 $
$\frac{a c(a-c)^2}{(a+c)^2}>0 \text { can be true. }$