Question

$\alpha ,\beta$ are the roots of the equation $K\left(x^2-x\right)+x+5=0$. If $K_1\&K_2$ are the two values of $K$ for which the roots $\alpha ,\beta$ are connected by the relation $(\alpha /\beta )+(\beta /\alpha )=4/5$. Find the value of $\left(K_1/K_2\right)+\left(K_2/K_1\right)$

Answer 1

Answer: 254

$K\left(x^2-x\right)+x+5=0;k{x}^2-x(k-1)+5=0$
$\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{4}{5}\Rightarrow \frac{{\alpha }^2+{\beta }^2}{\alpha \beta }=\frac{4}{5}\Rightarrow 5\left({\alpha }^2+{\beta }^2\right)=4\alpha \beta$
$5\left[(\alpha +\beta {)}^2-2\alpha \beta \right]=4\alpha \beta$
$5\left[\frac{(K-1{)}^2}{K^2}-\frac{2\times 5}{K}\right]=\frac{4\times 5}{K}\Rightarrow K^2+1-2K-10K=4K$

$\frac{K_1}{K_2}+\frac{K_2}{K_1}=\frac{{\left(K_1+K_2\right)}^2}{K_1{K}_2}-2$
$=256-2$
$=254$