Let $a$ be a positive constant. When we move the graph of the quadratic function $y = \frac{1}{4}x^2$ by parallel translation, the resulting parabola and the $x$-axis intersect at $(-2a, 0)$ and $(4a, 0)$. Let us consider the equation $y = f(x)$ of this parabola.
(1) The function $f(x)$ can be expressed as $$f(x) = \frac{\mathbf{A}}{\mathbf{B}}(x - \mathbf{C}a)(x + \mathbf{D}a)$$
(2) The range of values of $x$ such that the value of $y = f(x)$ is less than or equal to $10a^2$ can be obtained by solving the inequality $$x^2 - \mathbf{E}ax - \mathbf{FG}a^2 \leqq 0,$$ and it is $-\mathbf{H}a \leqq x \leqq \mathbf{I}a$.
(3) Suppose that the length of the segment between the intersections of the straight line $y = 10a$ and the graph of $y = f(x)$ is 10. Since $\mathbf{J}\sqrt{\mathbf{K}}a^2 + \mathbf{LM}a = 10$, we see that the value of $a$ is $\frac{\mathbf{N}}{\mathbf{O}}$.

Let $a$ be a positive constant. When we move the graph of the quadratic function $y = \frac{1}{4}x^2$ by parallel translation, the resulting parabola and the $x$-axis intersect at $(-2a, 0)$ and $(4a, 0)$. Let us consider the equation $y = f(x)$ of this parabola.
(1) The function $f(x)$ can be expressed as $$f(x) = \frac{\mathbf{A}}{\mathbf{B}}(x - \mathbf{C}a)(x + \mathbf{D}a)$$
(2) The range of values of $x$ such that the value of $y = f(x)$ is less than or equal to $10a^2$ can be obtained by solving the inequality $$x^2 - \mathbf{E}ax - \mathbf{FG}a^2 \leqq 0,$$ and it is $-\mathbf{H}a \leqq x \leqq \mathbf{I}a$.
(3) Suppose that the length of the segment between the intersections of the straight line $y = 10a$ and the graph of $y = f(x)$ is 10. Since $\mathbf{J}\sqrt{\mathbf{K}}a^2 + \mathbf{LM}a = 10$, we see that the value of $a$ is $\frac{\mathbf{N}}{\mathbf{O}}$.