If the line y = x - 1 bisects two chords of the parabola y^ 2 = 4bx...

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