Question

If the sum of the roots of quadratic equation $a{x}^2+bx+c=0$ is equal to the sum of the squares of their reciprocal then
$\frac{a}{c},\frac{b}{a},\frac{c}{b},$ are in

• A. G.P.
• B. H.P.
• C. A.G.P.
• D. A.P.

Answer 1

The correct option is B H.P.

It is given that

$\alpha +\beta =\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$

$=\frac{(\alpha +\beta {)}^2-2\alpha .\beta }{{\alpha }^2.{\beta }^2}$

$\frac{-b}{a}=\frac{b^2-2ac}{c^2}$

$-b{c}^2=a{b}^2-2a^{2}c$

$2a^{2}c=a{b}^2+b{c}^2$

Dividing the entire equation by abc, we get

$\frac{2a}{b}=\frac{b}{c}+\frac{c}{a}$

Hence

$\frac{b}{c},\frac{a}{b}\frac{c}{a}$ are in A.P.

Hence their reciprocals will be in H.P.