Question

If the curve $y=a{x}^2+2bx+c(a,b,c\in R,a,c\ne 0)$ never meets the $x$-axis, then $a,b,c$ can be in

• A. A.P.
• B. G.P.
• C. H.P.
• D. none

Answer 1

Answer: (C)

$D<0\Rightarrow 4b^2-4ac<0$
$b^2-ac<0$
(i) If $b=\frac{a+c}{2}\Rightarrow (a+c{)}^2-4ac<0\Rightarrow (a-c{)}^2<0$
(ii)$b^2=ac$
$\therefore 0<0$
(iii)$b=\frac{2ac}{a+c}$
$\therefore \frac{4a^2{c}^2}{(a+c{)}^2}-ac<0$
$\frac{ac}{(a+c{)}^2}\left[4ac-(a+c{)}^2\right]<0$
$\frac{ac(a-c{)}^2}{(a+c{)}^2}>0\text{ can be true. }$