Explanation

(b)
Given equation is
$y^{2} - 8y - x + 19 = 0$
$\Rightarrow \ \ (y - 4)^{2} = x - 19 + 16$
$\ \Rightarrow \ (y - 4)^{2} = (x - 3)$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ Y^{2} = 4AX$
Where, $Y = \ y - 4,\ A = \frac{1}{4}\$ and $X = x - 3\$
$\therefore\ Focus = (A,\ 0) = \ \left( \frac{1}{4},\ 0 \right) = \left( \frac{13}{4},\ 4 \right)$
Vertex$= (3,4)$
Directrix $X = - \frac{1}{4}$
$\Rightarrow \ \ x - 3 = - \frac{1}{4}$
$\Rightarrow \ x = \frac{11}{4}$