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

$\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$
$\Rightarrow \frac{2x-4-x+1}{(x-1)(x-2)}=\frac{2}{k}$
$\Rightarrow 2x^2-6x+4=k(x-3)$
$\Rightarrow 2x^2-x(6+k)+(4+3k)=0$
This equation has no solution then
$(6+k{)}^2<4\cdot 2(4+3k)$
$\Rightarrow k^2-12k+4<0$
$\Rightarrow k\in (6-4\sqrt{2},6+4\sqrt{2})$
$\Rightarrow k=1,2,3,...,11$
Sum of all values of $k=$ $11\left(\frac{11+1}{2}\right)=66$