If the two lines x + a - 1 y = 1 and 2 x + a 2 y = 1 , a ∈ R - 0,1 are perpendicular, then the distance of their point of intersection from the origin is

Step-by-step solution

If two lines a 1 x + b 1 y = c 1 and a 2 x + b 2 y = c 2 are perpendicular, then a 1 a 2 + b 1 b 2 = 0 . Since, given lines x + a - 1 y = 1 and 2 x + a 2 y = 1 are perpendicular, hence, we get 1 ⋅ 2 + a - 1 ⋅ a 2 = 0 ⇒ a 3 - a 2 + 2 = 0 ⇒ a + 1 ⋅ a 2 - 2 a + 2 = 0 ⇒ a = - 1 or a 2 - 2 a + 2 = 0 But, the discriminant of the quadratic equation is - 2 2 - 4 × 1 × 2 = 4 - 8 = - 4 , hence, no real roots exist. Thus, the only possible value of a is - 1 . And, hence the lines