Question

If $p,q,r$ in harmonic progression and $p$ & $r$ be different having same sign, then the roots of the equation $p{x}^2+qx+r=0$ are

• A. real and equal.
• B. real and distinct.
• C. irrational.
• D. imaginary.

Answer 1

Answer: (D)

imaginary.

$p,q,r$ in harmonic progression and $p,r$ be different having same sign.
$\Rightarrow q=\frac{2pr}{p+r}$ and $pr>0$
Now, discriminant of $p{x}^2+qx+r=0$
$D=q^2-4pr={\left(\frac{2pr}{p+r}\right)}^2-4pr$
$=4pr\left(\frac{-p^2-r^2-pr}{(p+r{)}^2}\right)$
$=-4pr\left(\frac{p^2+r^2+pr}{(p+r{)}^2}\right)<0$
$\Rightarrow D<0$
$\therefore$ Roots are imaginary.
Hence, option D.