For $f_1,f_2\in\Bbb R[x,y]$, let $S=\{k\in\Bbb R\mid kf_1+(1-k)f_2$ is not constant and reducible in $\Bbb R[x,y]\}$.
It is easy to see $\#S$ can be $0,1,2,3$:
$f_1=1,f_2=1$, then $S=\varnothing$, $\#S=0$.
$f_1=xy,f_2=1$, then $S=\{1\}$, $\#S=1$.
$f_1=x^2-1,f_2=y^2-1$, then $S=\{0,1\}$, $\#S=2$.
$f_1=x^2-y^2,f_2=y^2-1$, then $S=\{0,\frac12,1\}$, $\#S=3$.
Can $\#S$ be other finite number?