Question

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

Answer 1

Answer: 66

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