Question

lf $p,q,r$ are in H.P. and $p$ and $r$ are having same sign, then the roots of the equation $p{x}^2+qx+r=0$ will be

• A. Real
• B. Equal
• C. Imaginary
• D. Real and equal

Answer 1

The correct option is C Imaginary

$D=b^2-4ac=q^2-4pr$
$p,q,r$ are in H.P.
$\Rightarrow \frac{p+r}{pr}=\frac{2}{q}$
$q=\frac{2pr}{p+r}$
$q^2-4pr=\frac{4p^2{r}^2}{(p+r{)}^2}-4pr.$
$=4pr(pr-(p+r{)}^2)$
$=4pr(-(p^2+r^2+pr\left)\right)$as $p$ and $r$ are of same sign .
$q^2-4pr=-4pr(p^2+r^2+pr)\le 0.$
$\therefore$ Roots are imaginary .