If the line y = x - 1 bisects two chords of the parabola y 2 = 4 b x which are passing through the point b , - 2 b , then the length of the latus rectum can be equal to

Step-by-step solution

Any point on y = x – 1 can be taken as h , h – 1 Taking h , h - 1 as the midpoint of chord, the equation of chord is y h - 1 - 4 b x + h 2 = h - 1 2 - 4 b h Which passes through the point b , - 2 b ⇒ - 2 b h - 1 - 2 b b + h = h 2 - 2 h + 1 - 4 b h ⇒ h 2 - 2 h + 2 b 2 - 2 b + 1 = 0 As, above equation has two real roots, hence, - 2 2 - 4 1 2 b 2 - 2 b + 1 > 0 ⇒ 2 b 2 - 2 b < 0 ⇒ b < 1 ⇒ 4 b < 4 ⇒ length of latus rectum is less than 4