Question

If $a\ne b\ne c$ & are in H.P then roots of the equation $a{x}^2+2bx+c=0$ are

• A. Pure imaginary
• B. Pure real
• C. Two non-real roots whose sum and product is real
• D. None of these

Answer 1

The correct option is C Two non-real roots whose sum and product is real

Given that, $a{x}^2+2bx+c=0$

Since, $a,b,c$ are in H.P

Therefore, $b$ is the Harmonic mean of $a$ & $c$

$\Rightarrow H.M.=b$ ...(1)

and Geometric mean of $a$ and $c$ is $\sqrt{ac}$

$\Rightarrow G.M.=\sqrt{ac}$ ...(2)
Consider $D=4b^2-4ac=4(b^2-ac)$

$\Rightarrow D={(H.M.)}^2-{(G.M.)}^2$ ...[ From (1) & (2) ]

Since, $H.M.<G.M.$

Therefore, $D<0$

Therefore, the roots are imaginary.

Now roots are imaginary & conjugate of each other, we know that $z+\overline{z}$ and $z\overline{z}$ is pure real.