Any point on $y=x–1$ can be taken as $\left(h,h–1\right)$
Taking $\left(h,h-1\right)$ as the midpoint of chord, the equation of chord is $y\left(h-1\right)-4b\left(\frac{x+h}{2}\right)={\left(h-1\right)}^2-4bh$
Which passes through the point $\left(b,-2b\right)$
$\Rightarrow$ $-2b\left(h-1\right)-2b\left(b+h\right)=h^2-2h+1-4bh$
$\Rightarrow$ $h^2-2h+\left(2b^2-2b+1\right)=0$
As, above equation has two real roots, hence, 
${\left(-2\right)}^2-4\left(1\right)\left(2b^2-2b+1\right)>0$
$\Rightarrow$ $2b^2-2b<0\Rightarrow b<1\Rightarrow 4b<4$
$\Rightarrow$ length of latus rectum is less than $4$