Question

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

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

Answer 1

Answer: (A)

$A.P$

Let $\alpha$ and $\beta$ are the roots of $a{x}^2+bx+c=0$

We know $\alpha +\beta =\frac{-b}{a}$ $,\alpha .\beta =\frac{c}{a}$

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

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

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

Put values of $\alpha$ and $\beta$, we get

$\frac{-b}{a}=\frac{{\left(\frac{-b}{a}\right)}^2-2\frac{c}{a}}{{\left(\frac{c}{a}\right)}^2}$

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

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

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

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

Therefore, $a{b}^2,a^{2}c,b{c}^2$ are in $A.P.$

Option A is correct.