Let y = f x represent a parabola with focus - 1 2 , 0 and directrix y = - 1 2 . Then S = x ∈ ℝ : tan - 1 f x + sin - 1 f x + 1 = π 2 :

Step-by-step solution

Equation of parabola with focus ( - 1 2 ,   0 ) and directrix y = - 1 2 is: x + 1 2 2 = y + 1 4 ⇒ y = x 2 + x Put f ( x ) = y = x + x 2 in S , tan - 1 x x + 1 + sin - 1 x 2 + x + 1 = π 2 ⇒ 0 ≤ x 2 + x + 1 ≤ 1 ⇒ x 2 + x ≤ 0             … 1 Also, x 2 + x ≥ 0                       … 2 By 1   &   2 , we have x 2 + x = 0 ⇒ x = 0 , - 1 S contains 2 elemen