Question

The sum of all integral values of $k ( k \neq 0)$ for which the equation $\frac{2}{ x -1}-\frac{1}{ x -2}=\frac{2}{ k }$ in $x$ has no real roots, is ______.

Answer 1

$ \frac{2}{ x -1}-\frac{1}{ x -2}=\frac{2}{ k }$
$ x \in R -\{1,2\} $
$\Rightarrow k (2 x -4- x +1)=2\left( x ^{2}-3 x +2\right) $
$ \Rightarrow k ( x -3)=2\left( x ^{2}-3 x +2\right) $
for $ x \neq 3, \,\,\, k =2\left( x -3+\frac{2}{ x -3}+3\right) $
$x-3+\frac{2}{x-3} \geq 2 \sqrt{2}, \forall x>3$
$\&\,x-3+\frac{2}{x-3} \leq-2 \sqrt{2}, \forall x<-3$
$\Rightarrow 2\left(x-3+\frac{2}{x-3}+3\right) \in(-\infty, 6-4 \sqrt{2}] \cup[6+4 \sqrt{2}, \infty)$
for no real roots
$k \in(6-4 \sqrt{2}, 6+4 \sqrt{2})-\{0\}$
Integral $k \in\{1,2 \ldots \ldots 11\}$
Sum of $k=66$