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

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

$\frac{ K _1}{ K _2}+\frac{ K _2}{ K _1}=\frac{\left( K _1+ K _2\right)^2}{ K _1 K _2}-2$
$=256-2$
$ = 254 $