The complete set of non-zero values of k such that the equation  $|x^2-10x+9|=kx$  is satisfied by atleast one and atmost three values of  x, is

For the given condition to be satisfied, the value of k must lie between the slopes of line OA and OB 
For slope of line OA,
$-(x^2-10x+9)=kx$  must have equal roots.
$\Rightarrow {(k-10)}^2-36=0\Rightarrow k=4,16\Rightarrow k=4$
For slope of the line OB,                                                  

$x^2-10x+9=kx must have equal roots$

$$\begin{gathered} \Rightarrow {(k+10)}^2-36=0\Rightarrow k=-4,-16 \\ \therefore k\in [4,\infty )\cup (-\infty ,-16].] \end{gathered}$$