Question

If the sum of the roots of the equation $a{x}^2+bx+c=0$ is equal to the sum of the square of their reciprocals. Then ${bc}^2,{ca}^2,{ab}^2$ are in

• A. A.P.
• B. H.P.
• C. G.P.
• D. None of these.

Answer 1

The correct option is A A.P.

Let $\alpha ,\beta$ be the roots of $a{x}^2+bx+c=0$
Then, $\alpha +\beta =-\frac{b}{a}$ and $\alpha \beta =\frac{c}{a}$
Now,
$\alpha +\beta =\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}\Rightarrow (-\frac{b}{a})=\frac{{\alpha }^2+{\beta }^2-\alpha \beta }{{\alpha }^2{\beta }^2}\Rightarrow -\frac{b}{a}=\frac{\frac{b^2}{a^2}-\frac{c}{a}}{\frac{c^2}{a^2}}\Rightarrow -\frac{b{c}^2}{a}=b^2-ca\Rightarrow -b{c}^2=b^{2}a-c{a}^2\Rightarrow b^{2}a+b{c}^2=c{a}^2$
Hence, ${bc}^2,{ca}^2,{ab}^2$ are in A.P.