Question

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

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

Answer 1

Answer: (A)

AP

Consider the given equation.

$a{x}^2+bx+c=0$ ……….. (1)

Let $\alpha$ and $\beta$ are the roots of this equation.

So,

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

$\alpha \beta =\frac{c}{a}$

According to the question,

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

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

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

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

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

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

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

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

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

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