Explanation

(b)
Let the equation of line be $y = mx + c.$If this line touch the parabola $y^{2} = 8x$, then
$y = mx + \frac{2}{m}$
This line also touches the circle $x^{2} + y^{2} = 2$, then radius = perpendicular distance from centre (0,0) to the line
$\Rightarrow \ \ \sqrt{2} = \left| \frac{0 + 0 - \frac{2}{m}}{\sqrt{1 + m^{2}}} \right|$
$\Rightarrow \ m^{2}\left( 1 + m^{2} \right) = 2\ \Rightarrow \ m = 1$
$\therefore$ This required equation of tangent be
$y = x + 2$