Explanation

(a)
Given parametric equation of parabola is
$x = t^{2} + 1,\ y = 2t + 1$
$\Rightarrow x = \left( \frac{y - 1}{2} \right)^{2} + 1$
$\Rightarrow {(y - 1)}^{2} = 4(x - 1)$
$\Rightarrow Y^{2} = 4X$
Vertex is (1, 1), length of latusrectum $= 4$
Clearly, equation of directrix is
$X = - 1\ \Rightarrow x - 1 = - 1\ \Rightarrow x = 0$