The parabola (y + 1) 2 = a(x-2) passes through the point (1, -2). The equation of its directrix is

Given (y+1)^2 = a(x-2) passes through (1,-2): substituting gives (-2+1)^2 = a(1-2) → 1 = -a → a = -1, so the curve is (y+1)^2 = -(x-2). Comparing to standard form (Y)^2 = 4pX with X = x-2, Y = y+1: 4p = -1, so p = -1/4, and vertex (X=0,Y=0) corresponds to (x,y) = (2,-1). For this form, the directrix is X = -p, i.e., x - 2 = 1/4, so x = 9/4, giving the directrix equation 4x - 9 = 0. This is confirmed independently and matches the flag's own working. None of the four listed options (4x+1=0, 4x-1=0, 4x+q=0, 4x-q=0) equals 4x-9=0, and testing the DB's marked answer (4x+1=0) against the point (1,-2) shows it is inconsistent with any valid value of 'a'. The correct directrix (4x-9=0) simply isn't among the given choices.

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