Explanation

(a)
Given equation of curve is
$y = a(e^{\frac{x}{a}} + e^{- x/a})$
$\Rightarrow \ \ \ \ \frac{dy}{dx} = a\left( e^{\frac{x}{a}}.\frac{1}{a} - e^{\frac{- x}{a}}.\frac{1}{a} \right)$
Since, the tangent is parallel to $x$-axis,
$\therefore\ \ \ \ \ \ \frac{dy}{dx} = 0$
$\Rightarrow \ \ \ \ \ \ \left( e^{\frac{x}{a}} - e^{\frac{- x}{a}} \right) = 0$
$\Rightarrow \ \ \ \ \ \ \ e^{2x/a} = 1\ \ \ \ \ \Rightarrow \ \ \ \ \ \ x = 0$