Let $x_1=:x$ and $(x_2, \ldots, x_n)=:y\in{\mathbb R}^{n-1}$. The the Minkowski sum $M$ of the unit ball and the given ellipsoid is rotationally symmetric with respect to the $x$-axis and consists of all points $z=(x'+x'', y'+y'') \in {\mathbb R}^n$ with $${x'^2\over a^2}+|y'|^2\leq 1\ ,\qquad x''^2 +|y''|^2\leq 1\ .$$ A hyperplane $x={\rm const.}$ intersects $M$ in an $(n-1)$-dimensional ball whose radius can be found by maximizing $$|y'+y''|\leq |y'|+|y''|\leq \sqrt{1-x'^2/a^2} + \sqrt{1-x''^2}$$ under the constraint $x'+x''=x$.
Let $x_1=:x$ and $(x_2, \ldots, x_n)=:y\in{\mathbb R}^{n-1}$. The the Minkowski sum $M$ of the unit ball and the given ellipsoid is rotationally symmetric with respect to the $x$-axis and consists of all points $z=(x'+x'', y'+y'') \in {\mathbb R}^n$ with $${x'^2\over a^2}+|y'|^2\leq 1\ ,\qquad x''^2 +|y''|^2\leq 1\ .$$ A hyperplane $x={\rm const.}$ intersects $M$ in an $(n-1)$-dimensional ball whose radius can be found by maximizing $$|y'+y''|\leq |y'|+|y''|\leq \sqrt{1-x'^2/a^2} + \sqrt{1-x''^2}$$ under the constraint $x'+x''=x$.