Question

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

• A. AP
• B. GP
• C. HP
• D. none of these

Answer 1

Answer: (A)

AP

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

Then $\beta +\alpha =-\frac{b}{a};\alpha \beta =\frac{c}{a}$

Now

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

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

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

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