Let $a \in (-\infty, -1]$ , $c \in \mathbb R_+$ and $d \in (0,1)$ be constant real numbers. We define the parabola : $$ f(x) := ax^2 + c$$ For all $m \in \mathbb R$, we also define the geometric line $y = t_m(x)$ that is tangent to the parabola in the point $(m, f(m))$ : $$ \begin{align}t_m(x) &:= f(m)+f'(m)\left(x-m\right) \\\\ &= (am^2+c) + 2am(x-m) \\\\ &= (2am)x + (c-am^2) \end{align} $$
For each point $T_m = (m, f(m))$ , we let $Q_m$ be a point such that:
- $\Vert \overrightarrow{T_mQ_m}\Vert$ = d
- $\overrightarrow{T_mQ_m} \perp t_m(x)$
- $Q_m$, is closer to the $(Ox$ and $(Oy$ axes than $T_m$ (or more informally stated, $Q_m$ is selected so that it lays "inside" the parabola)
(it is guaranteed that $Q_m$ exists for any $m \in \mathbb R$)
Let $g = \{ Q_k : k \in \mathbb R\}$ be a function represented as a collection of these points.
Plotting the function $g$ indicates that the function is continuous and even, presenting geometric tangent lines $y = \tau_m(x)$ which are always parallel to the respective tangents $y = t_m(x)$ of $f$, for all $m \in \mathbb R$.
I am interested to know:
- Can $g(x)$ can be exactly given by a mathematical expression? (it is a parabola / a hyperbola / etc.)
- Given $\alpha \in \mathbb R$ such that $g(\alpha) = 0$, what is the area between the geometric elements $x=-\alpha$, $x=\alpha$, $y=0$ and $y=g(x)$? (that is, the integral of $g(x)$ above the ($Ox$ axis and between the zeroes of the function $g$).
Thank you very much.